Question

Determine the zero(s) of the quadratic function y = 6x^ + 13x + 5. Show all steps.

Answer 1

The quadratic function is y = 6x^2 + 13x + 5.

To find the zero(s) of the quadratic function, we set y equal to zero and solve for x:

0 = 6x^2 + 13x + 5

To factor this quadratic equation, we need to find two numbers whose sum is 13 and whose product is (6)(5) = 30. The numbers are 10 and 3, since 10 + 3 = 13 and 10 * 3 = 30.

0 = 6x^2 + 10x + 3x + 5 (applying the numbers 10 and 3 as the coefficients of x)
0 = (6x^2 + 10x) + (3x + 5)

Now, we factor the equation by grouping:

0 = 2x(3x + 5) + 1(3x + 5)
0 = (2x + 1)(3x + 5)

Now, we can set each factor equal to zero:

2x + 1 = 0 or 3x + 5 = 0

Solving the first equation, we get:

2x = -1
x = -1/2

Solving the second equation, we get:

3x = -5
x = -5/3

Therefore, the zeros of the quadratic function y = 6x^2 + 13x + 5 are x = -1/2 and x = -5/3.