 # 7.1 – Completing the Square

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7.1 – Completing the Square
Many quadratic equations can not be solved by factoring. Other techniques are required to solve them. Examples: x2 = 20 5x = 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 5x = 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

The Derivation of the Quadratic Formula

The Derivation of 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.

Example: solve by factoring

Example: solve by 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− 𝑥−3 −6=0 𝑥=2 2 2 − −3 −6=0 −6=0 0=0 𝑥=− 3 2 2 − 3 2 − − 3 2 −3 −6=0 −3− −3−3 −6=0 − −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 − =0 − =0 2 2 −5 2 +6=0 0=0 𝑡=1 − =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 − =0 16−13(4)+36=0 16−52+36=0 0=0 𝑥=±3 −3 4 −13 − =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 − −7 − −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 −7 − 1 4 −2=0 −2=0 0=0