10.5 Completing the Square

10.5 – Completing the Square Goals / “I can…”  Solve quadratic equations by completing the square

10.5 – Completing the Square Review: 3  Remember we’ve solved quadratics using 3 different ways: Graphing Square Roots Factoring

y = x 2 – 4x – 5 Solutions are -1 and – Completing the Square How many solutions are there? What are they?

1. 25x 2 = 16 ANSWER –, 2. 9m 2 = 100 ANSWER 10 3 –, b = 0 ANSWER no solution 10.5 – Completing the Square Use the Square Root method to solve:

Example 1 x 2 – 2x – 24 = 0 (x + 4)(x – 6) = 0 x + 4 = 0 x – 6 = 0 x = –4 x = 6 Example 2 x 2 – 8x + 11 = 0 x 2 – 8x + 11 is prime; therefore, another method must be used to solve this equation – Completing the Square

The easiest trinomials to look at are often perfect squares because they always have the SAME characteristics.

10.5 – Completing the Square x + 8x + 16 is factored into (x + 4) notice that the 4 is (½ * 8) 2 2 2

10.5 – Completing the Square ALWAYS This is ALWAYS the case with perfect squares. The last term in the binomial can be found by the formula ½ b Using this idea, we can make polynomials that aren’t perfect squares into perfect squares. 2

10.5 – Completing the Square Example: x + 22x + ____ What number would fit in the last term to make it a perfect square? 2

10.5 – Completing the Square (½ * 22) = 121 SO….. x + 22x should be a perfect square. (x + 11) 2 2 2

10.5 – Completing the Square What numbers should be added to each equation to complete the square? x + 20x x - 8x x + 50x 2 2 2

This method will work to solve ALL quadratic equations; HOWEVER it is “messy” to solve quadratic equations by completing the square if a aa a ≠ 1 and/or b is an odd number. Completing the square is a G GG GREAT choice for solving quadratic equations if a = 1 and b is an even number – Completing the Square

Example 1 a = 1, b is even x 2 – 6x - 7 = 0 x 2 – 6x + 9 = (x – 3) 2 = 16 x – 3 = ± 4 x = 7 OR 1 Example 2 a ≠ 1, b is not even 3x 2 – 5x + 2 = 0 OR x = 1 OR x = ⅔ 10.5 – Completing the Square

Solving x + bx = c x + 8x = 48 I want to solve this using perfect squares. How can I make the left side of the equation a perfect square? 2 2

10.5 – Completing the Square Use ½ b (½ * 8) = 16 MUST Add 16 to both sides of the equation. (we MUST keep the equation equivalent) x + 8x + 16 = Make the left side a perfect square binomial. (x + 4) =

10.5 – Completing the Square x + 4 = 8 SO………. x + 4 = 8x + 4 = -8 x = 4x =

10.5 – Completing the Square Solving x + bx + c = 0 x + 12x + 11 = 0Since it is not a perfect square, move the 11 to the other side. x + 12x = -11 Now, can you complete the square on the left side? 2 2 2

c Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 1. x 2 + 8x + cANSWER16; (x + 4) 2 2. x 2  12x + c 3. x 2 + 3x + c ANSWER 36; (x  6) 2 ANSWER ; (x  ) – Completing the Square

Solve x 2 – 16x = –15 by completing the square. SOLUTION Write original equation. x 2 – 16x = –15 Add, or (– 8) 2, to each side. – x 2 – 16x + (– 8) 2 = –15 + (– 8) 2 Write left side as the square of a binomial. (x – 8) 2 = –15 + (– 8) 2 Simplify the right side. (x – 8) 2 = – Completing the Square

Take square roots of each side. x – 8 = ±7 Add 8 to each side. x = 8 ± 7 ANSWER The solutions of the equation are = 15 and 8 – 7 = – Completing the Square

x + 12x + ? = ? x + 12x + = (x + ) = 2 2 2

10.5 – Completing the Square Complete the square x - 20x + 32 = 0 2

10.5 – Completing the Square Complete the square x + 3x – 5 = 0 2

10.5 – Completing the Square Complete the square x + 9x = 136 2

10.5 – Completing the Square Still a little foggy? If so, watch this video to see if it will helpthis video