Many quadratic equations can not be solved by factoring. Other techniques are required to solve them. 7.1 – Completing the Square x 2 = 20 5x = 0 Examples: ( x + 2) 2 = 18 ( 3x – 1) 2 = – 4 x 2 + 8x = 1 2x 2 – 2x + 7 = 0

If b is a real number and if a 2 = b, then a = ± √¯‾. 20 Square Root Property b x 2 = 20 x = ± √‾‾ x = ± √‾‾‾‾ 4·5 x = ± 2 √‾ 5 – 11 5x = 0 x = ± √‾‾‾ 5x 2 = – 55 x 2 = – 11 x = ± i √‾‾‾ – Completing the Square

If b is a real number and if a 2 = b, then a = ± √¯‾. 18 Square Root Property b ( x + 2) 2 = 18 x + 2 = ± √‾‾ x + 2 = ± √‾‾‾‾ 9·2 x +2 = ± 3 √‾ 2 x = – 2 ± 3 √‾ 2 –4–4 ( 3x – 1) 2 = – 4 3x – 1 = ± √‾‾ 3x – 1 = ± 2i 3x = 1 ± 2i 7.1 – Completing the Square

Review: ( x + 3) 2 x 2 + 2(3x) + 9 x 2 + 6x x 2 + 6x + 9 ( x + 3) ( x + 3) 2 x 2 – 14x x 2 – 14x + 49 ( x – 7) ( x – 7) – Completing the Square

Examples x 2 + 9xx 2 – 5x 7.1 – Completing the Square

x 2 + 8x = – Completing the Square Example

5x 2 – 10x + 2 = 0 5x 2 – 10x = – 2 or Example 7.1 – Completing the Square

2x 2 – 2x + 7 = 0 2x 2 – 2x = – 7 or Example 7.1 – Completing the Square

The quadratic formula is used to solve any quadratic equation. The quadratic formula is: Standard form of a quadratic equation is: 7.2 – The Quadratic Formula The Quadratic Formula

The Derivation of the Quadratic Formula 7.2 – The Quadratic Formula

The Derivation of the Quadratic Formula 7.2 – The Quadratic Formula

State the values of a, b, and c from each quadratic equation. The quadratic formula is: Standard form of a quadratic equation is: 7.2 – The Quadratic Formula

Example: solve by factoring 7.2 – The Quadratic Formula

Example: solve by the quadratic formula 7.2 – The Quadratic Formula

Example 7.2 – The Quadratic Formula

Example 7.2 – The Quadratic Formula