7.1 – Completing the Square Many quadratic equations can not be solved by factoring. Other techniques are required to solve them. Examples: x2 = 20 5x2 + 55 = 0 ( x + 2)2 = 18 ( 3x – 1)2 = –4 x2 + 8x = 1 2x2 – 2x + 7 = 0

7.1 – Completing the Square Square Root Property If b is a real number and if a2 = b, then a = ±√¯‾. b x2 = 20 5x2 + 55 = 0 x = ±√‾‾ 20 5x2 = –55 x = ±√‾‾‾‾ 4·5 x2 = –11 x = ± 2√‾ 5 x = ±√‾‾‾ –11 x = ± i√‾‾‾ 11

7.1 – Completing the Square Square Root Property If b is a real number and if a2 = b, then a = ±√¯‾. b ( x + 2)2 = 18 ( 3x – 1)2 = –4 x + 2 = ±√‾‾ 18 3x – 1 = ±√‾‾ –4 x + 2 = ±√‾‾‾‾ 9·2 3x – 1 = ± 2i x +2 = ± 3√‾ 2 3x = 1 ± 2i x = –2 ± 3√‾ 2

7.1 – Completing the Square Review: ( x + 3)2 x2 – 14x x2 + 2(3x) + 9 x2 + 6x + 9 x2 – 14x + 49 x2 + 6x ( x – 7) ( x – 7) ( x – 7)2 x2 + 6x + 9 ( x + 3) ( x + 3) ( x + 3)2

7.1 – Completing the Square Examples x2 + 9x x2 – 5x

7.1 – Completing the Square Example x2 + 8x = 1 x2 + 8x = 1

7.1 – Completing the Square Example 2x2 – 2x + 7 = 0 2x2 – 2x = –7

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

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

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

7.4 – More Equations There are additional techniques to solve other types of equations. Example:                   𝑥=− 3 2            

7.4 – More Equations   Check: 2𝑥−3 2 +5 2𝑥−3 −6=0 𝑥=2 2𝑥−3 2 +5 2𝑥−3 −6=0  𝑥=2 2 2 −3 2 +5 2 2 −3 −6=0 1 2 +5 1 −6=0 0=0 𝑥=− 3 2  2 − 3 2 −3 2 +5 2 − 3 2 −3 −6=0 −3−3 2 +5 −3−3 −6=0 −6 2 +5 −6 −6=0 0=0

7.4 – More Equations Example:                                

7.4 – More Equations   Check: 2𝑡+1 2 −5 2𝑡+1 +6=0 𝑡= 1 2 2𝑡+1 2 −5 2𝑡+1 +6=0 𝑡= 1 2  2 1 2 +1 2 −5 2 1 2 +1 +6=0 1+1 2 −5 1+1 +6=0 2 2 −5 2 +6=0 0=0  𝑡=1 2 1 +1 2 −5 2 1 +1 +6=0 3 2 −5 3 +6=0 0=0

7.4 – More Equations Example: 𝑥 4 −13 𝑥 2 +36=0 𝐿𝑒𝑡 𝑦= 𝑥 2 𝑎𝑛𝑑 𝑦 2 = 𝑥 4 𝑥 2 =4 𝑥 2 =9 𝑦 2 −13𝑦+36=0 𝑥=± 4 𝑥=± 9 𝑦−4 𝑦−9 =0 𝑥=±2 𝑥=±3 𝑦−4=0 𝑦−9=0 𝑦=4 𝑦=9 𝑥 2 =4 𝑥 2 =9

7.4 – More Equations   Check: 𝑥 4 −13 𝑥 2 +36=0 𝑥=±2 −2 4 −13 −2 2 +36=0 16−13(4)+36=0 16−52+36=0 0=0 𝑥=±3  −3 4 −13 −3 2 +36=0 81−13(9)+36=0 81−117+36=0 0=0

7.4 – More Equations Example: 4 𝑥 4 −7𝑥 2 −2=0 𝐿𝑒𝑡 𝑦= 𝑥 2 𝑎𝑛𝑑 𝑦 2 = 𝑥 4 𝑥 2 =− 1 4 4 𝑦 2 −7𝑦−2=0 𝑥 2 =2 𝑦−2 4𝑦+1 =0 𝑥=± 2 𝑥=±𝑖 1 4 𝑦−2=0 4𝑦+1=0 𝑦=− 1 4 𝑦=2 𝑥=± 1 2 𝑖 𝑥 2 =− 1 4 𝑥 2 =2

7.4 – More Equations   Check: 4 𝑥 4 −7𝑥 2 −2=0 𝑥=± 2 𝑥=± 2  − 2 4 −7 − 2 2 −2=0 16−7 2 −2=0 0=0 𝑥=± 1 2 𝑖  4 − 1 2 𝑖 4 −7 − 1 2 𝑖 2 −2=0 4 1 16 −7 − 1 4 −2=0 1 4 + 7 4 −2=0 0=0