Completing the Square Be ready to grade the homework! ALGEBRA – LESSON 118 Completing the Square Be ready to grade the homework!

Completing the Square We know that there is a difference between factoring an equation and solving an equation. Once we have factored an equation, we can solve it by setting each binomial equation to zero and finding and answer for each “brick”. We have always used zero for the right side, but it doesn’t have to be a zero. We also learned in Lesson 96 that if x2 = 9, then we solve by taking the square root of both sides, taking into account that the answer could be positive or negative. x2 = 9 becomes x = ±3.

Completing the Square Sometimes we will have to force an equation to become something we can factor. Remember, we can do anything we want to an equation as long as we do it to both sides. See if you can find a pattern as we factor here: x2 + 6x + 9 (x + 3)(x + 3) (x + 3)2 x2 + 10x + 25 (x + 5)(x + 5) (x + 5)2 x2 + 8x + 16 (x + 4)(x + 4) (x + 4)2 These all become binomials that are squared – or perfect squares. AND, the number in the “brick” is half of the 2nd term.

Completing the Square To solve these equations, we could use a method called “completing the square”. 1) Rewrite the equation to be x2 + bx = c x2 + 2x = 63 2) Find half of b and then square it. Add that number to both sides of the equation. x2 + 2x + 1 = 63 + 1 3) Factor the left side and combine terms on the right side. (x + 1)(x + 1) = 64 (x + 1)2= 64 4) Find the square root of each side. Ö(x + 1)2= Ö64 5) Solve. x + 1= ± 8 x = 7 or -9

Completing the Square After rearranging, what number will we add to both sides? x2 − 4x − 96 = 0 22 = 4 x2 − 8x + 20 = 5 42 = 16 n2 + 12n + 10 = −10 62 = 36 5n2 + 20n + 10 = −5 102 = 100 n2 − 18n − 83 = 5 92 = 81 x2 + 2x − 35 = 0 12 = 1 x2 − 1x + 30 = 6 .52 = .25

Completing the Square Solve by completing the square. x2 + 4x = 21 Half of 4 is 2. 22 = 4. x2 + 4x + 4 = 21 + 4 (x + 2)(x + 2) = 25 (x + 2)2= 25 Ö(x + 2)2= Ö25 x + 2= ± 5 x = 3 or -7 #2

Completing the Square Solve by completing the square. x2 – 6x = 27 Half of 6 is 3. 32 = 9. x2 – 6x + 9 = 27 + 9 (x - 3)(x - 3) = 36 (x - 3)2= 36 Ö(x - 3)2= Ö36 x – 3 = ± 6 x = 9 or -3 #3

Completing the Square Solve by completing the square. x2 + x = 20 Half of 1 is .5 .52 = .25 x2 + x + .25 = 20 + .25 (x + .5)(x + .5) = 20.25 (x + .5)2= 20.25 Ö(x + .5)2= Ö20.25 x + .5 = ± 4.5 x = 4 or -5 #4

Completing the Square Solve by completing the square. x2 - x = 20 Half of 1 is .5 .52 = .25 x2 - x + .25 = 20 + .25 (x - .5)(x - .5) = 20.25 (x - .5)2= 20.25 Ö(x - .5)2= Ö20.25 x - .5 = ± 4.5 x = 5 or -4 #5

Homework: Worksheet