This will all give you the equation (x-1)^2=16. Because you´re taking this value away from the constant, you will add it to the other side of the equation (this might not make sense at first, but if the constant were on the variable side, you would be subtracting). In this case, you will add 1 because it perfectly factors out into (x-1)^2. Next, you want to add a value to the variable side so that when you factor that side, you will have a perfect square. Because there is a 3 in front of x^2, you will divide both sides by 3 to get x^2-2x=5. Next, you want to get rid of the coefficient before x^2 (a) because it won´t always be a perfect square. To do this, you will subtract 8 from both sides to get 3x^2-6x=15. To complete the square, first, you want to get the constant (c) on one side of the equation, and the variable(s) on the other side. We offer tutoring programs for students in K-12, AP classes, and college.You don´t need another video because I´m about to explain it to you! Say you have the equation 3x^2-6x+8=23. SchoolTutoring Academy is the premier educational services company for K-12 and college students. Interested in algebra tutoring services? Learn more about how we are assisting thousands of students each academic year. The “missing piece” that is completed is a square that has both sides equal to b/2.įigure 4: Another way to look at completing the square. The equation x 2 + 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. Geometric Representation of Completing the SquareĪnother way of looking at completing the square is by looking at the geometric representation of how a square is completed. If the square root of x is negative, then x + 3 = -1, so x = -4.įigure 3: The path to completing a square. If the square root of x is positive, then x + 3 = 1, so x = -2. 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. Factoring the left side of the equation, (x + 3) 2 = 1. Using the addition rules of algebra, 9 can be added to both sides of the equation to form x 2 + 6x +9 = -8 +9. The b coefficient in this equation is 6, and half of 6 is 3. If the equation were a perfect square trinomial, half of the b coefficient would be squared to form the c coefficient. This turns x 2 + 6x + 8 = 0 to x 2 + 6x = -8, by subtracting 8 from both sides. 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. That property, as well as addition rules in algebra, can be used to complete the square. In the perfect square trinomial x 2 + 8x + 16, for example, half of 8 is 4, and 4 2 is 16. When the coefficient for the x 2 term is 1, half of the b coefficient (for the x term) is squared to form the c coefficient. It is not a perfect square trinomial like x 2 + 8x + 16. Suppose the equation were x 2 + 6x + 8 = 0. Similarly, 100 – 80 + 16 = 36.įigure 2: Use the definition of square roots to solve perfect square trinomials. The solutions for x would still be 2 and -10, because if x = 2, then 4 + 16 + 16 = 36. Suppose that the perfect square trinomial were expanded so that x 2 + 8x + 16 = 36. If 6 is positive, then x equals 2, but if 6 is negative, then x = -10. Then, by the definition of square roots, x + 4 ± 6. The simplest way to solve them is by using the definition of square roots. Then x ± √75/3, or x ±5.įigure 1: Solving for x when x is squaredĮquations in the form (x + c) 2 = d are also quadratic equations in the form of perfect square trinomials. If the coefficient a is greater than 1, then the value of x will equal ±√p/a. An easier way in symbol form is that √144 = ☑2. Using the definition of square roots, there are 2 square roots of 144, 12, and -12, because 12∙12 = 144, and -12∙-12 also equals 144. For example, if x 2 = 144, then x = 12, the square root of 144. The value of x will equal the square root of p, or √p, using the radical). Suppose the coefficient a is equal to 1, so that the equation is x 2 = p (and p is greater than 0, so that the problem has a real-number solution). Equations in the Form ax 2 = pĮquations in the form ax 2 = p are quadratic equations. Many quadratic equations can be solved by a process called “completing the square.” The process uses the definitions of square roots, as well as the principles of adding or subtracting constants.
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