Algebra 1 Notes SOL A.4 Solving Equations by Completing the Square Mrs. Grieser Name: ______Date: ______Block: ______“Completing the Square”  Remember when we looked at polynomials that factored to a  b2 ?

a b2  a ba b a2  2abb2

 Example: x 32  x2  6x 9

 Take an expression of the form x 2  bx and find the third term (the “c”) that will form a “perfect square trinomial,” one that can be factored by squaring a binomial.

o Create perfect square trinomials…

x2 + 6x + ______= (x + ______)2 x2 + 10x + ______= (x + ______)2

x2 + x + ______= (x + ______)2 x2 - 11x + ______= (x - ______)2

 We call this process “completing the square”

 Basic rule: find the b coefficient and half it then square it to find c

Process: Start with x 2  bx c Example 1 : x 2  8x  c Example 2 : x 2  5x  c b b 5 1) Identify b. Take half of b  . 1) b = 8 = 4 1) b = 5 = 2 2 2

2 2  b   5  25 2) Square   . This is our c! 2) = 42 = 16 = c 2) =   = = c  2   2  4

2 3) Write and factor the polynomial created: 3) x 2  8x 16  x  4 25 5 2 2 2 2    b   b  3) x  5x    x   x2  bx   =  x   4  2   2   2 

You try: Complete the square (find c that will form a perfect square trinomial)… a) x 2 12x  c b) x 2  4x  c c) x 2  3x  c

Algebra 1 Notes SOL A.4 Solving Equations by Completing the Square Mrs. Grieser Page 2 Solving Quadratic Equations by Completing the Square

Use completing the square to solve quadratic equations of the form x 2  bx  d

Process: Start with x 2  bx  d Example 3: x 2 16x  15 2 2 2  b   b   16 1) Identify b value. Take half of b and square it    1)   =   = 64  2   2   2  2) Add to both sides of the equation: 2) x 2 16x  64  15 64 = 2 2  b   b  2 x 2  bx   = d    x 16x  64  49  2   2 

3) Factor the polynomial created, take the square root of 3) x  82  49 b both sides, and subtract : 2 x 8  7 2 2  b   b  x  78  x   = d     2   2  Solutions: x = 15, 1

2 2  b   b  x + =  d    x =  d    -  2   2 

Example 4: Solve 2x 2  20x  8  0 by completing the square.

1) Put in the form (note that you need to divide through by a): ______

2) Complete the square on the left side of the equation; add to BOTH sides of equation:

______

3) Take the square root of both sides, and subtract to isolate x: ______

4) The solutions are: ______

Other examples: Solve by completing the square… a) x 2  6x  5  0 b) x 2  2x 1 c) 4x 2  4x  3

You try: Solve the equations by completing the square. Round to the nearest hundredth if necessary… a) x 2  2x  3 b) m2 10m  8 c) 3g 2  24g  27  0