Overview
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.
Equations in the Form ax2 = p
Equations in the form ax2 = p are quadratic equations. Suppose the coefficient a is equal to 1, so that the equation is x2 = 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 √p, using the radical). For example, if x2 = 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∙12 = 144, and -12∙-12 also equals 144. An easier way in symbol form is that √144 = ±12. If the coefficient a is greater than 1, then the value of x will equal ±√p/a. For example, suppose 3x2 = 75. Then x ± √75/3, or x ±5.
Figure 1: Solving for x when x is squared
Equations in the Form (x + c)2 = d
Equations in the form (x + c)2 = 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 = 36. Then, by the definition of square roots, x + 4 ± 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 + 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.
Figure 2: Use the definition of square roots to solve perfect square trinomials.
Completing the Square
Suppose the equation were x2 + 6x + 8 = 0. It is not a perfect square trinomial like x2 + 8x + 16. When the coefficient for the x2 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 + 8x + 16, for example, half of 8 is 4, and 42 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 + 6x + 8 = 0 to x2 + 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 or 9. Using the addition rules of algebra, 9 can be added to both sides of the equation to form x2 + 6x +9 = -8 +9. Factoring the left side of the equation, (x + 3)2 = 1. Therefore, x +3 =±√1. 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. 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.
Figure 3: The path to completing a square.
Geometric Representation of Completing the Square
Another way of looking at completing the square is by looking at the geometric representation of how a square is completed. The equation x2 + 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 “missing piece” that is completed is a square that has both sides equal to b/2.
Figure 4: Another way to look at completing the square.
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