QuestionFind the zeros of the function $f(x)=7 x^{2}+54 x-16$ . Enter your answers as a comma-separated list: $x=$

Studdy Solution

STEP 1

Assumptions1. The function is a quadratic function of the form $f(x) = ax^ + bx + c$ . . We are asked to find the zeros of the function, which are the x-values for which $f(x) =0$ .

STEP 2

We can find the zeros of the function by setting the function equal to zero and solving for $x$ . This gives us the equation $7x^2 +54x -16 =0$

STEP 3

This is a quadratic equation, which can be solved using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 -ac}}{2a}$

STEP 4

We can identify the coefficients $a$ , $b$ , and $c$ from the equation $7x^2 +54x -16 =0$ as follows $a =7$ , $b =54$ , and $c = -16$ .

STEP 5

Substitute the coefficients $a$ , $b$ , and $c$ into the quadratic formula to solve for $x$ $x = \frac{-54 \pm \sqrt{54^2 -47(-16)}}{2*7}$

STEP 6

implify the expression under the square root $x = \frac{-54 \pm \sqrt{2916 +448}}{14}$

STEP 7

Further simplify the expression under the square root $x = \frac{-54 \pm \sqrt{3364}}{14}$

STEP 8

Take the square root $x = \frac{-54 \pm58}{14}$

STEP 9

olve for $x$ by using the plus and minus signs separately $x = \frac{-54 +58}{14}$ $x2 = \frac{-54 -58}{14}$

STEP 10

Calculate the values of $x$ and $x2$ $x = \frac{4}{14} =0.2857$ $x2 = \frac{-112}{14} = -8$ The zeros of the function $f(x) =7x^2 +54x -16$ are $x =0.2857, -8$ .

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