select the approxiate value of x that is the solution to f(x)=0 where f(x)=-9x^2+6x+7

Find the appropriate value(s) of x that is/are the solution(s) to f(x)=0 where f(x)=-9x^2+6x+7.

The values of x that make f(x) = 0 are called the Zeros of f(x). The Zeros are also the x-intercepts if they are real. If the Zeros are Imaginary then f(x) has no x-intercepts.

So we want to solve the Quadratic Equation 0=-9x^2+6x+7 for the x-values that make it true, its Zeros, a.k.a., Roots.

Let's use the Quadratic Formula to find the Zeros:

If 0 = ax^2 + bx + c, then

x = (- b ± sqrt(b^2-4ac))/(2a)

Identify parameters:

a = -9, the coefficient of x^2

b = 6, the coefficient of x

c = 7, the constant

Substitute:

x = (- 6 ± sqrt(6^2-4(-9)(7)))/(2(-9))

Simplify:

x = (- 6 ± sqrt(36+36(7)))/(-18)

x = (- 6 ± sqrt(36(1+7)))/(-18)

x = (- 6 ± sqrt(36*4*2))/(-18)

x = (- 6 ± 6*2*sqrt(2))/(-6*3)

x = (1 ± 2*sqrt(2))/3

x = (1/3) ± (2/3)*sqrt(2)