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 b2 ? a b2 a ba b a2 2abb2 Example: x 32 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 b 16 1) Identify b value. Take half of b and square it 1) = = 64 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 82 49 both sides, and subtract : x 8 7 2 b x 78 = d 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 theb left side of the equation; add to BOTH sides of equation: 2 __________________________________ 3) Take the square root of both sides, and subtract to isolate x: ______________________ 2 b 4) The solutions are: __________________ 2 Other examples: Solve by completing the square… a) x 2 6x 5 0 b) x 2 2x 1 c) 4x 2 4x 3 2 b x 2 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 .