Equations in the form ax2<\/sup> = p are quadratic equations. Suppose the coefficient a is equal to 1, so that the equation is x2<\/sup> = p (and p is greater than 0, so that the problem has a real-number solution). The value of x will equal the square root of p, or \u221ap, using the radical). For example, if x2<\/sup> = 144, then x = 12, the square root of 144. Using the definition of square roots, there are 2 square roots of 144, 12, and -12, because 12\u221912 = 144, and -12\u2219-12 also equals 144. An easier way in symbol form is that \u221a144 = \u00b112. If the coefficient a is greater than 1, then the value of x will equal \u00b1\u221ap\/a. For example, suppose 3x2<\/sup> = 75. Then x \u00b1 \u221a75\/3, or x \u00b15.<\/p>\n Figure 1: Solving for x when x is squared<\/p>\n Equations in the form (x + c)2<\/sup> = d are also quadratic equations in the form of perfect square trinomials. The simplest way to solve them is by using the definition of square roots. Suppose that (x + 4)2<\/sup> = 36. Then, by the definition of square roots, x + 4 \u00b1 6. If 6 is positive, then x equals 2, but if 6 is negative, then x = -10. Suppose that the perfect square trinomial were expanded so that x2<\/sup> + 8x + 16 = 36. The solutions for x would still be 2 and -10, because if x = 2, then 4 + 16 + 16 = 36. Similarly, 100 – 80 + 16 = 36.<\/p>\n Figure 2: Use the definition of square roots to solve perfect square trinomials.<\/p>\n Suppose the equation were x2<\/sup> + 6x + 8 = 0. It is not a perfect square trinomial like x2<\/sup> + 8x + 16. When the coefficient for the x2<\/sup> term is 1, half of the b coefficient (for the x term) is squared to form the c coefficient. In the perfect square trinomial x2<\/sup> + 8x + 16, for example, half of 8 is 4, and 42<\/sup> is 16. That property, as well as addition rules in algebra, can be used to complete the square. The first step is to turn the equation into a form that can be used to complete the square by moving the c term to the other side of the equation. This turns x2 <\/sup> + 6x + 8 = 0 to x2<\/sup> + 6x = -8, by subtracting 8 from both sides. If the equation were a perfect square trinomial, half of the b coefficient would be squared to form the c coefficient. The b coefficient in this equation is 6, and half of 6 is 3. The c term is 32<\/sup> or 9. Using the addition rules of algebra, 9 can be added to both <\/strong>sides of the equation to form x2<\/sup> + 6x +9 <\/strong>= -8 +9<\/strong>. Factoring the left side of the equation, (x + 3)2<\/sup> = 1. Therefore, x +3 =\u00b1\u221a1. It is very important to remember that a number has a positive and a negative square root in order to solve both values of x.<\/strong> If the square root of x is positive, then x + 3 = 1, so x = -2. If the square root of x is negative, then x + 3 = -1, so x = -4.<\/p>\n Figure 3: The path to completing a square.<\/p>\n Another way of looking at completing the square is by looking at the geometric representation of how a square is completed. The equation x2<\/sup> + bx = a (which is the same equation as when the c term is moved), can be seen as a geometric figure that consists of a square with sides x and two rectangles with sides bx. The \u201cmissing piece\u201d that is completed is a square that has both sides equal to b\/2.<\/p>\n Figure 4: Another way to look at completing the square.<\/p>\n Interested in algebra tutoring services<\/a>? Learn more about how we are assisting thousands of students each academic year.<\/p>\n SchoolTutoring Academy<\/span> is the premier educational services company for K-12 and college students. We offer tutoring programs for\u00a0 students in K-12, AP classes, and college. To learn more about how we help parents and students in Nanaimo, BC, Canada: visit Tutoring in Nanaimo, BC, Canada<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" Overview Many quadratic equations can be solved by a process called \u201ccompleting the square.\u201d The process uses the definitions of square roots, as well as the principles of adding or subtracting constants. Equations in the Form ax2 = p Equations in the form ax2 = p are quadratic equations. Suppose the coefficient a is equal […]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[339,1716],"class_list":["post-7157","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-completing-the-square","tag-square-roots"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/7157","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=7157"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/7157\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=7157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=7157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=7157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"monohybridcross\"](\"https:\/\/schooltutoring.com\/help\/wp-content\/uploads\/sites\/2\/2014\/06\/Solving-equations.png\")
<\/a><\/p>\nEquations in the Form (x + c)2<\/sup> = d<\/h3>\n
<\/a><\/p>\nCompleting the Square<\/h3>\n
<\/a><\/p>\nGeometric Representation of Completing the Square<\/h3>\n
<\/a><\/p>\n