{"id":2511,"date":"2026-09-15T08:55:09","date_gmt":"2026-09-15T08:55:09","guid":{"rendered":"https:\/\/www.think10x.ai\/blog\/?p=2511"},"modified":"2026-09-15T08:55:09","modified_gmt":"2026-09-15T08:55:09","slug":"difference-of-squares","status":"publish","type":"post","link":"https:\/\/www.think10x.ai\/blog\/difference-of-squares\/","title":{"rendered":"Difference of Squares Formula and Examples"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"2511\" class=\"elementor elementor-2511\" data-elementor-post-type=\"post\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5886ff3 e-flex e-con-boxed e-con e-parent\" data-id=\"5886ff3\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-156ce9a elementor-widget elementor-widget-text-editor\" data-id=\"156ce9a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The difference of squares formula states that a\u00b2 \u2212 b\u00b2 = (a + b)(a \u2212 b). It lets you factor any expression that is one perfect square subtracted from another into two binomials, one with a sum and one with a difference. This difference of squares formula appears constantly in algebra, from simplifying expressions to solving quadratic equations, and recognizing the pattern on sight is one of the fastest ways to cut through a factoring problem.<\/span><\/p><h2>Key Takeaways<\/h2><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The formula is a\u00b2 \u2212 b\u00b2 = (a + b)(a \u2212 b), and it works only when you are subtracting one perfect square from another.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">A sum of squares (a\u00b2 + b\u00b2) cannot be factored using this identity over the real numbers.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The difference of squares identity can be proved in one step by expanding (a + b)(a \u2212 b) and watching the middle terms cancel.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Always check for a greatest common factor (GCF) before applying the rule, since pulling out the GCF first can reveal a hidden difference of squares underneath.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The same pattern extends to expressions with higher powers, such as x\u2074 \u2212 16, which factors as (x\u00b2 + 4)(x\u00b2 \u2212 4) and then again as (x\u00b2 + 4)(x + 2)(x \u2212 2).<\/span><\/li><\/ul><h2>What Is the Difference of Squares?<\/h2><p><span style=\"font-weight: 400;\">The difference of two squares is any algebraic expression where one squared term is subtracted from another, written in the general form a\u00b2 \u2212 b\u00b2. According to <\/span><a href=\"https:\/\/tutorial.math.lamar.edu\/classes\/alg\/factoring.aspx\" rel=\"noopener\"><span style=\"font-weight: 400;\">Lamar University&#8217;s algebra course <\/span><\/a><span style=\"font-weight: 400;\">on factoring polynomials, recognizing the difference of squares as a special factoring pattern is one of the most useful shortcuts in algebra, since it appears inside more complex problems like solving quadratics, simplifying rational expressions, and reducing polynomial fractions.<\/span><\/p><p><span style=\"font-weight: 400;\">The word &#8220;difference&#8221; here means subtraction, and &#8220;squares&#8221; means each term is raised to the power of 2. So x\u00b2 \u2212 9 qualifies (x squared minus 3 squared), but x\u00b2 + 9 does not, because there is no subtraction.<\/span><\/p><p><b>Read more &#8211;<\/b><a href=\"https:\/\/www.think10x.ai\/blog\/ai-explainer-video-tool\/\">\u00a0<span style=\"font-weight: 400;\">See how Think10x.ai explains algebra topics step by step<\/span><\/a><span style=\"font-weight: 400;\"> with narrated video walkthroughs.<\/span><\/p><h2>The Difference of Squares Formula<\/h2><p><span style=\"font-weight: 400;\">The difference of squares formula is <\/span><b>a\u00b2 \u2212 b\u00b2 = (a + b)(a \u2212 b)<\/b><\/p><p><span style=\"font-weight: 400;\">This tells you that any expression in the form &#8220;first term squared minus second term squared&#8221; can be rewritten as the product of two brackets: one containing the sum of the two terms, and one containing the difference.<\/span><\/p><h3>Proof of the difference of squares identity<\/h3><p><span style=\"font-weight: 400;\">The difference of squares identity is easy to prove by expanding the right side:<\/span><\/p><p><span style=\"font-weight: 400;\">(a + b)(a \u2212 b) = a(a \u2212 b) + b(a \u2212 b) = a\u00b2 \u2212 ab + ab \u2212 b\u00b2 = a\u00b2 \u2212 b\u00b2<\/span><\/p><p><span style=\"font-weight: 400;\">The two middle terms, \u2212ab and +ab, cancel each other out, leaving only a\u00b2 \u2212 b\u00b2. This cancellation is why the identity works, and it also explains why a sum of squares (a\u00b2 + b\u00b2) has no equivalent factoring over real numbers.<\/span><\/p><h2>The Difference of Squares Rule: How to Recognize It<\/h2><p><span style=\"font-weight: 400;\">The difference of squares rule has three conditions that must all be true before you can apply the formula:<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The expression has exactly two terms (it is a binomial).<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Both terms are perfect squares, meaning each one is the result of squaring a whole number, variable, or simpler expression.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The two terms are connected by subtraction, not addition.<\/span><\/li><\/ol><p><span style=\"font-weight: 400;\"><br \/>If all three conditions are met, you can factor directly using a\u00b2 \u2212 b\u00b2 = (a + b)(a \u2212 b). If any one condition fails, the identity does not apply.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-01e7655 e-flex e-con-boxed e-con e-parent\" data-id=\"01e7655\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-67b0fee eael-table-align-center eael-dt-th-align-left elementor-widget elementor-widget-eael-data-table\" data-id=\"67b0fee\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-data-table.default\">\n\t\t\t\t\t\t\t<div class=\"eael-data-table-wrap\" data-table_id=\"67b0fee\" id=\"eael-data-table-wrapper-67b0fee\" data-custom_responsive=\"false\">\n\t\t\t<table class=\"tablesorter eael-data-table center\" id=\"eael-data-table-67b0fee\">\n\t\t\t    <thead>\n\t\t\t        <tr class=\"table-header\">\n\t\t\t\t\t\t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Expression<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Is it a difference of squares?<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Why \/ Why not<\/span><\/th>\n\t\t\t        \t\t\t\t        <\/tr>\n\t\t\t    <\/thead>\n\t\t\t  \t<tbody>\n\t\t\t\t\t\t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\u00b2 \u2212 25\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tYes\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\u00b2 and 25 are both perfect squares, connected by subtraction\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t4x\u00b2 \u2212 9y\u00b2\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tYes\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t4x\u00b2 = (2x)\u00b2 and 9y\u00b2 = (3y)\u00b2, subtracted\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\u00b2 + 16\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tNo\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tIt uses addition, not subtraction\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\u00b2 \u2212 7\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tNo (over integers)\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t7 is not a perfect square integer\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\u00b3 \u2212 8\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tNo\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\u00b3 is not a square; this is a difference of cubes\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t    <\/tbody>\n\t\t\t<\/table>\n\t\t<\/div>\n\t  \t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-d008657 e-flex e-con-boxed e-con e-parent\" data-id=\"d008657\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-d30e5bc elementor-widget elementor-widget-text-editor\" data-id=\"d30e5bc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h2>How to Use the Difference of Squares<\/h2><p><span style=\"font-weight: 400;\">How to use difference of squares in four steps<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Check that the expression is a binomial with subtraction.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Confirm both terms are perfect squares by identifying what each one is the square of.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Write two brackets: one with the sum of the square roots, one with the difference.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Check your answer by expanding the brackets back out to confirm it matches the original.<\/span><\/li><\/ol><h3>Step-by-step walkthrough: Factor 49x\u00b2 \u2212 64<\/h3><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Is it a binomial with subtraction? Yes.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Is 49x\u00b2 a perfect square? Yes, (7x)\u00b2 = 49x\u00b2.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Is 64 a perfect square? Yes, 8\u00b2 = 64.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Write the brackets: (7x + 8)(7x \u2212 8).<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Check: (7x + 8)(7x \u2212 8) = 49x\u00b2 \u2212 56x + 56x \u2212 64 = 49x\u00b2 \u2212 64. Correct.<\/span><\/li><\/ul><p><b>Learn more &#8211;<\/b><a href=\"https:\/\/www.think10x.ai\/studio\"> <span style=\"font-weight: 400;\">Upload any factoring problem to Think10x.ai<\/span><\/a><span style=\"font-weight: 400;\"> and get a narrated video explanation showing each step.<\/span><\/p><h2>How to Solve Difference of Squares Equations<\/h2><p><span style=\"font-weight: 400;\">How to solve the difference of squares equations means setting the factored form equal to zero and solving each bracket separately.<\/span><\/p><h3>Example 1. <span style=\"font-weight: 400;\">Solve x\u00b2 \u2212 36 = 0<\/span><\/h3><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Factor -&gt; x\u00b2 \u2212 36 = (x + 6)(x \u2212 6) = 0.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Set each bracket to zero -&gt; x + 6 = 0 gives x = \u22126, and x \u2212 6 = 0 gives x = 6.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The solutions are x = 6 and x = \u22126.<\/span><\/li><\/ol><p><span style=\"font-weight: 400;\"><br \/>This is one of the most common applications of the difference of squares method in algebra exams, and the factoring step is often the quickest route to the answer compared to using the quadratic formula on the same equation.<\/span><\/p><h2>Difference of Squares Examples<\/h2><p><span style=\"font-weight: 400;\">Here are different examples covering the types that show up most often in homework and exams.<\/span><\/p><h3>Example 1. Basic factoring<\/h3><p><span style=\"font-weight: 400;\">Factor x\u00b2 \u2212 81.<\/span><\/p><p><span style=\"font-weight: 400;\">Both terms are perfect squares: x\u00b2 = (x)\u00b2 and 81 = (9)\u00b2. Using the difference of two squares formula: x\u00b2 \u2212 81 = (x + 9)(x \u2212 9).<\/span><\/p><h3>Example 2. Coefficients greater than 1<\/h3><p><span style=\"font-weight: 400;\">Factor 25a\u00b2 \u2212 4b\u00b2.<\/span><\/p><p><span style=\"font-weight: 400;\">25a\u00b2 = (5a)\u00b2 and 4b\u00b2 = (2b)\u00b2. So 25a\u00b2 \u2212 4b\u00b2 = (5a + 2b)(5a \u2212 2b).<\/span><\/p><h3>Example 3. Factoring out a GCF first<\/h3><p><span style=\"font-weight: 400;\">Factor 3x\u00b2 \u2212 75.<\/span><\/p><p><span style=\"font-weight: 400;\">The two terms share a common factor of 3. Pull it out first: 3(x\u00b2 \u2212 25). Now x\u00b2 \u2212 25 is a difference of squares: 3(x + 5)(x \u2212 5).<\/span><\/p><h3>Example 4. Nested difference of squares<\/h3><p><span style=\"font-weight: 400;\">Factor x\u2074 \u2212 16.<\/span><\/p><p><span style=\"font-weight: 400;\">x\u2074 = (x\u00b2)\u00b2 and 16 = (4)\u00b2, so x\u2074 \u2212 16 = (x\u00b2 + 4)(x\u00b2 \u2212 4). But x\u00b2 \u2212 4 is itself a difference of squares: x\u00b2 \u2212 4 = (x + 2)(x \u2212 2). The fully factored form is (x\u00b2 + 4)(x + 2)(x \u2212 2).<\/span><\/p><h3>Example 5. Mental math shortcut<\/h3><p><span style=\"font-weight: 400;\">Calculate 47 \u00d7 53 without a calculator.<\/span><\/p><p><span style=\"font-weight: 400;\">Notice that 47 = 50 \u2212 3 and 53 = 50 + 3. So 47 \u00d7 53 = (50 \u2212 3)(50 + 3) = 50\u00b2 \u2212 3\u00b2 = 2500 \u2212 9 = 2491. This mental math trick, using the difference of squares method in reverse, works whenever two numbers are the same distance above and below a round number, and it is one of the clearest demonstrations of how to use the difference of squares outside a textbook.<\/span><\/p><p><b>Read more &#8211;<\/b><a href=\"https:\/\/www.think10x.ai\/blog\/math-solver-online-chatgpt-vs-think10x-ai\/\"> <span style=\"font-weight: 400;\">See how Think10x.ai&#8217;s solver handles algebra problems<\/span><\/a><span style=\"font-weight: 400;\"> compared to ChatGPT.<\/span><\/p><h2>Difference of Squares Problems<\/h2><p><span style=\"font-weight: 400;\">Try these difference of squares problems and check your solutions below.<\/span><\/p><p><b>Problem 1.<\/b><span style=\"font-weight: 400;\">\u00a0Factor 16x\u00b2 \u2212 49. Solution: (4x)\u00b2 \u2212 (7)\u00b2 = (4x + 7)(4x \u2212 7).<\/span><\/p><p><b>Problem 2.<\/b><span style=\"font-weight: 400;\">\u00a0Solve 9x\u00b2 \u2212 1 = 0. Solution: (3x + 1)(3x \u2212 1) = 0, so x = 1\/3 or x = \u22121\/3.<\/span><\/p><p><b>Problem 3.<\/b><span style=\"font-weight: 400;\">\u00a0Factor 2x\u00b3 \u2212 50x completely. Solution: Factor out 2x first: 2x(x\u00b2 \u2212 25) = 2x(x + 5)(x \u2212 5).<\/span><\/p><p><b>Problem 4.<\/b><span style=\"font-weight: 400;\">\u00a0Calculate 102 \u00d7 98 using the difference of squares. Solution: (100 + 2)(100 \u2212 2) = 100\u00b2 \u2212 2\u00b2 = 10000 \u2212 4 = 9996.<\/span><\/p><p><span style=\"font-weight: 400;\">If any step feels unclear,<\/span><a href=\"https:\/\/www.think10x.ai\/blog\/is-think10x-ai-the-best-chegg-alternative\/\"> <span style=\"font-weight: 400;\">a narrated video walkthrough of a similar problem<\/span><\/a><span style=\"font-weight: 400;\"> can show the full reasoning instead of just a static answer key.<\/span><\/p><p><b>Checklist before you submit an answer<\/b><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Did you check for a GCF before applying the formula?<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Are both terms actually perfect squares?<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Is the operation subtraction, not addition?<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Did you expand your answer back out to verify it matches the original expression?<\/span><\/li><\/ul><p><b>Learn more &#8211;<\/b><a href=\"https:\/\/www.think10x.ai\/for-students\">\u00a0<span style=\"font-weight: 400;\">Explore step-by-step walkthroughs for algebra and beyond<\/span><\/a><span style=\"font-weight: 400;\"> on Think10x.ai.<\/span><\/p><h2>How Think10x.ai Helps With Factoring<\/h2><p><span style=\"font-weight: 400;\">Factoring problems like the difference of squares follow a clear pattern, but the pattern only helps if you can spot it in the first place, especially when a GCF needs to come out first or when nested squares hide inside an expression like x\u2074 \u2212 16. Think10x.ai turns any factoring question, typed, spoken, or photographed from a worksheet, into a narrated video that walks through identification, factoring, and verification one step at a time.<\/span><\/p><p><span style=\"font-weight: 400;\">Students can pause mid-explanation and ask a follow-up, such as why 7 is not a perfect square or why (x\u00b2 + 4) cannot be factored further, and the platform picks up from that exact point.<\/span><a href=\"https:\/\/www.think10x.ai\/\"> <span style=\"font-weight: 400;\">Try it with your next algebra problem<\/span><\/a><span style=\"font-weight: 400;\"> and see the full walkthrough in under two minutes.<\/span><\/p><h2>Frequently Asked Questions<\/h2>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-68f8795 e-flex e-con-boxed e-con e-parent\" data-id=\"68f8795\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-6db0f95 elementor-widget elementor-widget-eael-adv-accordion\" data-id=\"6db0f95\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-adv-accordion.default\">\n\t\t\t\t\t            <div class=\"eael-adv-accordion\" id=\"eael-adv-accordion-6db0f95\" data-scroll-on-click=\"no\" data-scroll-speed=\"300\" data-accordion-id=\"6db0f95\" data-accordion-type=\"accordion\" data-toogle-speed=\"300\">\n            <div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"can-a-sum-of-squares-be-factored-the-same-way-\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"1\" aria-controls=\"elementor-tab-content-1151\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">Can a sum of squares be factored the same way? <\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-1151\" class=\"eael-accordion-content clearfix\" data-tab=\"1\" aria-labelledby=\"can-a-sum-of-squares-be-factored-the-same-way-\"><p><span style=\"font-weight: 400\">No. The identity a\u00b2 + b\u00b2 does not factor into real-number binomials, because expanding (a + b)(a + b) gives a\u00b2 + 2ab + b\u00b2, not a\u00b2 + b\u00b2. The middle term does not cancel the way it does with subtraction.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"what-if-the-expression-has-three-terms-instead-of-two-\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"2\" aria-controls=\"elementor-tab-content-1152\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">What if the expression has three terms instead of two? <\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-1152\" class=\"eael-accordion-content clearfix\" data-tab=\"2\" aria-labelledby=\"what-if-the-expression-has-three-terms-instead-of-two-\"><p><span style=\"font-weight: 400\">Then it is a trinomial, not a binomial, and the difference of squares formula does not apply directly. You would need to check whether it fits a different factoring pattern, such as a perfect square trinomial or a general quadratic.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"does-the-order-of-the-brackets-matter-\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"3\" aria-controls=\"elementor-tab-content-1153\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">Does the order of the brackets matter? <\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-1153\" class=\"eael-accordion-content clearfix\" data-tab=\"3\" aria-labelledby=\"does-the-order-of-the-brackets-matter-\"><p><span style=\"font-weight: 400\">No. (a + b)(a \u2212 b) and (a \u2212 b)(a + b) give the same result, since multiplication is commutative. The final factored answer is the same either way.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"how-do-i-know-when-to-use-the-difference-of-squares-versus-the-quadratic-formula-\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"4\" aria-controls=\"elementor-tab-content-1154\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">How do I know when to use the difference of squares versus the quadratic formula? <\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-1154\" class=\"eael-accordion-content clearfix\" data-tab=\"4\" aria-labelledby=\"how-do-i-know-when-to-use-the-difference-of-squares-versus-the-quadratic-formula-\"><p><span style=\"font-weight: 400\">If the expression is a binomial with two perfect squares separated by subtraction, the difference of squares is faster. If the expression is a full quadratic (three terms) or does not factor neatly, the quadratic formula is the more general tool.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>The difference of squares formula states that a\u00b2 \u2212 b\u00b2 = (a + b)(a \u2212 b). It lets you factor any expression that is one perfect square subtracted from another into two binomials, one with a sum and one with a difference. This difference of squares formula appears constantly in algebra, from simplifying expressions to [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":2532,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"https:\/\/media.mentomind.ai\/img\/t10x\/bp\/difference_of_squares_formulas.webp","fifu_image_alt":"","footnotes":""},"categories":[8,1],"tags":[],"class_list":["post-2511","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-blogs","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/2511","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/comments?post=2511"}],"version-history":[{"count":5,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/2511\/revisions"}],"predecessor-version":[{"id":2530,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/2511\/revisions\/2530"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/media\/2532"}],"wp:attachment":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/media?parent=2511"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/categories?post=2511"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/tags?post=2511"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}