Download presentation

Presentation is loading. Please wait.

2
**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

3
**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

4
**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

5
**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

6
**7.1 – Completing the Square**

Examples x2 + 9x x2 – 5x

7
**7.1 – Completing the Square**

Example x2 + 8x = 1 x2 + 8x = 1

8
**7.1 – Completing the Square**

Example 2x2 – 2x + 7 = 0 2x2 – 2x = –7

9
**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:

10
**7.2 – The Quadratic Formula**

The Derivation of the Quadratic Formula

11
**7.2 – The Quadratic Formula**

The Derivation of the Quadratic Formula

12
**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.

13
**7.2 – The Quadratic Formula**

Example: solve by factoring

14
**7.2 – The Quadratic Formula**

Example: solve by the quadratic formula

15
**7.2 – The Quadratic Formula**

Example

17
7.4 – More Equations There are additional techniques to solve other types of equations. Example: 𝑥=− 3 2

18
**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

19
7.4 – More Equations Example:

20
**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

21
**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

22
**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

23
**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

24
**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

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google