BELL-WORK Tell whether each expression is rational or irrational. +√144 Solve (3x – 8)2 = 169

HW 3.2(b) Due tomorrow: PW 9-5 # 7-18 (even) (round to the nearest hundredth where necessary)

HW 3.2(a) Solutions 5 OR -5 22 OR -22 16 OR -16 No solution 4 OR -4 3 3 3 OR -3 9 OR -9 7 OR -7 4 4 21. No solution

Guiding question: How are quadratics solved?

RECALL Quadratics can be solved by: Factoring and the ZPP Quadratic formula Graphing Square roots There is one more method that we will discuss! Quadratics can be solved by completing the square.

Solving Quadratics by Completing the Square (3x – 8)2 = 169 Notice that when we have a perfect square it is easy to solve the quadratic using square roots. Completing the square is a process that changes any quadratic trinomial to a perfect square trinomial. Once the trinomial has been converted to a perfect square then we can easily solve the quadratic by square roots.

Solving Quadratics by Completing the Square Example: Solve x2 + 20x + 36 = 0 by completing the square. This quadratic is not a perfect square. So let’s force it to be one. First ensure that the value of A is 1. Find half of B 10 and square it 100 Add and subtract this figure to the quadratic so that it stays balanced x2 + 20x +100 – 100 + 36 = 0 Notice that we have not changed the original quadratic.

Solving Quadratics by Completing the Square x2 + 20x +100 – 100 + 36 = 0 Consider x2 + 20x + 100 Factor it! = (x + 10)2 So x2 + 20x + 100 – 100 + 36 = 0 can be re-written as: (x + 10)2 – 100 + 36 = 0 (x + 10)2 – 64 = 0 (x + 10)2 = 64 x + 10 = + 8 x = -2 or x = -18

Solving Quadratics by Completing the Square This method will always work, but it is usually used when a quadratic trinomial is difficult to factor. Given Ax2 + Bx + C = 0, to complete the square: Step 1: Ensure that the value of A is 1. Step 2: Find half of B. Step 3: Add and subtract the square of this value. Step 4: Rewrite and solve the expression.

Solving Quadratics by Completing the Square Rationale of completing the square Remember the perfect square formula: (Ax + B)2 = Ax2 + 2ABx + B2

Solving Quadratics by Completing the Square Example: Solve x2 + 9x – 136 = 0 by completing the square. Step 1: The value of A is 1 Step 2: half of 9 is 9 2 squared 81 4 Step 3: x2 + 9x + 81 – 81– 136 = 0 4 4 Examine the first three terms.

Solving Quadratics by Completing the Square x2 + 9x + 81 4 = x2 + 2•1•9 + 2 = (x + 9/2)2 Step 4: So x2 + 9x + 81 – 81– 136 = 0 4 4 can be re-written as: (x + 9/2)2 – 81 – 136 = 0

Solving Quadratics by Completing the Square (x + 9/2)2 – 81 – 136 = 0 4 (x + 9/2)2 – 625 = 0 (x + 9/2)2 = 625 (x + 9/2) = + 25 2 x = -9 + 25 2 2 x = 8 OR -17

Did you get it? TB pg 564 # 13, 19

Did you get it? TB pg 564 # 13, 19 Check your answer using any other method of your choice.

Did you get it? TB pg 564 # 13, 19 Check your answer using any 2 other methods of your choice.

Who wants to answer the Guiding question? How are quadratics solved?

Every one needs a signature!!! Quiz 3.2 Review Every one needs a signature!!!