Solving Quadratic Equations by Extracting Square Roots.pptx

mathematics 9

1. Gabaldon Vocational Agriculture High School Solving Quadratic Equations by Extracting Square Roots September 04, 2023

2. Objectives familiarize numbers that are perfect squares; and solve quadratic equations by extracting square roots.

3. Chart of perfect Square

4. Quadratic equations that can be written in the form x 2 = k can be solved by applying the following properties : 1. If k > 0, then x 2 = k has two real solutions or roots: x =   2. If k = 0, then x 2 = k has one real solution or root: x = 0. 3. If k < 0, then x 2 = k has no real solutions or roots. The method of solving the quadratic equation x 2 =k is called extracting square roots.

5. Example 1 : Find the solutions of the equation x 2 -16 = 0 by extracting square roots. Step 1. Write the equation in the form x 2 = k. x 2 – 16 = 0 x 2 – 16+16 = 0 + 16 x 2 = 16 Step 2. Get the square roots of both sides x 2 = 16   =   x =  

6. To check, substitute these values in the original equation. For x = 4: x 2 – 16 = 0 4 2 – 16 = 0 16 – 16 = 0 0 = 0 For x = -4: x 2 – 16 = 0 -4 2 – 16 = 0 16 – 16 = 0 0 = 0

7. Both values of x satisfy the given equation. So the equation x 2 – 16 = 0 is true when x = 4 or when x = -4. Answer : The equation x 2 – 16 = 0 has two solutions: x = 4 or x = -4.

8. Example 2 : Solve the equation t 2 = 0 Step 1. Write the equation in the form x 2 = k. t 2 = 0 Step 2. Get the square roots of both sides t 2 = 0   =   t =0

9. Since t 2 equals 0, then the equation has only one solution. Answer : The equation t 2 = 0 has one solution: t = 0.

10. Example 3 : Find the roots of the equation s 2 + 9 = 0. Step 1. Write the equation in the form x 2 = k. s 2 +9 = 0 s 2 + 9 - 9 = 0 -9 s 2 = -9

11. Since -9 is less than 0, then the equation s 2 = -9 has no real solutions or roots . There is no real number when squared gives -9. Answer : The equation s 2 + 9 = 0 has no real solutions or roots.

12. Example 4 : Find the solutions of the equation ( x - 4) 2 - 25 = 0 by extracting square roots. Step 1. Write the equation in the form x 2 = k. (x-4) 2 – 25 = 0 Step 2. Get the square roots of both sides   =   x-4 = 5 (x-4) 2 – 25+25 = 0+25 (x-4) 2 = 25 (x-4) 2 = 25

13. x-4 = 5   Solve for x in the equation x-4 =   For x = 5 + 4 x = 9 For x = -5 + 4 x = -1

14. Let’s check the obtained values of x against the original equation For x = 9 (x-4) 2 -25 = 0 (9-4) 2 -25 = 0 (5) 2 -25 = 0 25-25 = 0 For x = 9 (x-4) 2 -25 = 0 (-1-4) 2 -25 = 0 (-5) 2 -25 = 0 25-25 = 0 Answer : The equation (x-4) 2 -25 = 0 has two solutions: x=9 or x = 1.

15. Let us try this… Find the solutions of equation 2(x-5) 2 = 32

16. Let us try this… Find the solutions of equation 4x 2 -225 = 0

17. Let us try this… Find the solutions of equation 3 (4x-1) 2 -1 = 11

18. Quiz # 2 Solve the following quadratic equations by extracting square roots. x 2 = 16 t 2 = 81 r 2 -100 = 0 x 2 -144 = 0 2s 2 = 50 (x-4) 2 = 169 (k+7) 2 =289 (2s-1) 2 = 225