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QuestionFind the zeros of the function f(x)=8x2+12x+3f(x)=8x^2+12x+3 using the quadratic formula. What are the x-intercepts?

Studdy Solution

STEP 1

Assumptions1. The function is a quadratic function of the form f(x)=ax+bx+cf(x) = ax^ + bx + c. . The zeros of the function are the x-values that make the function equal to zero.
3. The x-intercepts of the function are the points where the graph of the function crosses the x-axis, which are also the zeros of the function.
4. The quadratic formula is x=−b±b−4acax = \frac{-b \pm \sqrt{b^ -4ac}}{a}.

STEP 2

First, we need to identify the coefficients aa, bb, and cc in the quadratic function f(x)=8x2+12x+f(x) =8x^2 +12x +.
a=8,b=12,c=a =8, b =12, c =

STEP 3

Now, we will use the quadratic formula to find the zeros of the function.
x=−b±b2−ac2ax = \frac{-b \pm \sqrt{b^2 -ac}}{2a}

STEP 4

Substitute the values of aa, bb, and cc into the quadratic formula.
x=−12±122−4∗8∗32∗8x = \frac{-12 \pm \sqrt{12^2 -4*8*3}}{2*8}

STEP 5

implify the expression under the square root.
x=−12±144−9616x = \frac{-12 \pm \sqrt{144 -96}}{16}

STEP 6

Further simplify the expression under the square root.
x=−12±4816x = \frac{-12 \pm \sqrt{48}}{16}

STEP 7

implify the square root.
x=−12±4316x = \frac{-12 \pm4\sqrt{3}}{16}

STEP 8

implify the expression by dividing each term by4.
x=−3±34x = \frac{-3 \pm \sqrt{3}}{4}So, the zeros of the function are x=−3+34x = \frac{-3 + \sqrt{3}}{4} and x=−3−34x = \frac{-3 - \sqrt{3}}{4}.

STEP 9

The x-intercepts of the function are the same as the zeros of the function, since the x-intercepts are the x-values where the function equals zero.
So, the x-intercepts of the function are x=−3+34x = \frac{-3 + \sqrt{3}}{4} and x=−3−34x = \frac{-3 - \sqrt{3}}{4}.
The correct choice is A. The zeros and the x-intercepts are the same. They are x=−3+34x = \frac{-3 + \sqrt{3}}{4} and x=−3−34x = \frac{-3 - \sqrt{3}}{4}.

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