please help solve these

Question

find the zeros of the function by using the zero product property for the following:

 

4x^2-20x=0

3x^2+30x=0

-3x^2-15x+42=0

x^2-5x-14=0

9x^2-16=0

9x^2+66x-121=0

14x^2=55x-21

25x^2-81=0

x^2-32=-14x

-7x^2=34x-5

 

x^2-11x+18=0

 

14x^2-57x-27=0

 

25x^2-144=0

64x^2-48x+9=0

5x^2=-23x+42

x^2=6x=-55

 

 

 

solve with radical form (simplest

 

4x^2+5=41

5x^2-6=24

16X62+2=65

4(x-11)^2=36

64(x-5)^2=55

25(x+13+^2=45

4(x+2)^2-3=32

7x^2-14=28

 

4(x-3)^2=31

Answer 1

3x² - 30x = 0 ---> 3x(x - 10) = 0 ---> x₁ = 0 , x₂ = 10
x² - 5x - 14 = 0 ---> (x - 7)(x + 2) = 0 ---> x₁ = 7 , x₂ = -2 
9x² + 66x + 121 = 0 ---> (3x + 11)² = 0 ---> x = -11/3 
25x² - 81 = 0 ---> (5x - 9)(5x + 9) = 0 ---> x₁₂ = ±9/5
-7x² = 34x - 5 ---> 7x² + 34x - 5 = 0 ---> (7x - 1)(x + 5) = 0 ---> x₁=-5, x₂=1/7  
x² - 11x + 18 = 0 ---> (x - 2)(x - 9) = 0 ---> x₁ = 2 , x₂ = 9
14x² - 57x - 27 = 0 ---> (2x - 9)(7x + 3) = 0 ---> x₁ = 9/2 , x₂ = -3/7
25x² - 144 = 0 ---> (5x - 12)(5x + 12) = 0 ---> x₁₂ = ±12/5
5x² = -23x + 42 ---> 5x² + 23x - 42 = 0 ---> (5x - 7)(x + 6) = 0 ---> x₁ = -6 , x₂ = 7/5
4x² + 5 = 41 ---> 4x² + 5 - 41 = 0 ---> 4x² - 36 = 0 ---> 4(x² - 9) = 0 ---> x₁₂ = ± 3

Answer 2

First note that the zeros of a function are the values of x where the function equals 0. With that, the goal is to set the equation equal to 0 by moving all the terms to one side of the equation. After factoring, we can apply the zero product property, which states that if the product of two factors is zero then each factor is equal to zero, to solve for x. I will work out a few of the problems you've listed as examples:

1)   4x² - 20x = 0        factor out the greatest common factor ==>   4x

      (4x)(x - 5) = 0

     zero-product property ==>     4x = 0           and        x - 5 = 0

                                              4x/4 = 0/4                x - 5 + 5 = 0 + 5

                                                  x = 0                              x = 5

2)   -3x² - 15x + 42 = 0        factor out greatest common factor (-3) then divide both sides by it

     -3(x² + 5x - 14) = 0      ==>     x² + 5x - 14 = 0        factor the left hand side of equation

      (x + 7)(x - 2) = 0        ==>     x + 7 = 0        and        x - 2 = 0

                                                     x = -7           and         x = 2

3)   9x² - 16 = 0            notice that this is the difference of two squares

      (3x)² - (4)² = 0        ==>        (3x - 4)(3x + 4) = 0

      By the zero product property:      3x - 4 = 0          and          3x + 4 = 0

                                                        3x = 4                              3x = -4

                                                          x = 4/3                             x = -4/3

4)   x² - 32 = -14x          add 14x to both sides

      x² - 32 + 14x = 0      ==>     x² + 14x - 32 = 0        factor the left hand side of equation

     (x - 2)(x + 16) = 0       ==>     x - 2 = 0           and          x + 16 = 0

                                                   x = 2                                x = -16

5)   5x² - 6 = 24        subtract 24 from both sides then factor out greatest common factor (5)

      5x² - 30 = 0     ==>     5(x² - 6) = 0        divide both sides by 5

        x² - 6 = 0             add 6 to both sides then take the square root of both sides

        x² = 6      ==>      √(x²) = ± √(6)     ==>     x = ±√6

6)   4(x - 11)² = 36        divide both sides by 4

        (x - 11)² = 9          take the square root of both sides

        √(x - 11)² = ± √9     ==>     x - 11 = ± 3        add 11 to both sides

         x = 11 ± 3           ==>     x = 11 + 3      and     x = 11 - 3

                                               x = 14                      x = 8

7)   64(x - 5)² = 55        divide both sides by 64

      (x - 5)² = 55/64        take the square root of both sides

     √(x - 5)² = ± √(55/64) = ± √(55)/√(64)

      x - 5 = ± √(55)/8        add 5 to both sides to solve for x

      x = 5 ± √(55)/8