QuestionFind the zeros of the function $f(x)=4x^2+10x+5$ using the quadratic formula. What are the $x$ -intercepts?

Studdy Solution

STEP 1

Assumptions1. The function is a quadratic function given by $f(x)=4x^{}+10x+5$ . We are asked to find the zeros of the function using the quadratic formula3. The quadratic formula is given by $x=\frac{-b\pm\sqrt{b^{}-4ac}}{a}$ where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation in the form $ax^{}+bx+c=0$
4. The zeros of the function are the x-intercepts of the graph of the function

STEP 2

First, identify the coefficients $a$ , $b$ , and $c$ from the given quadratic function $f(x)=4x^{2}+10x+5$ .
$a=4$ $b=10$ $c=5$

STEP 3

Substitute the coefficients $a$ , $b$ , and $c$ into the quadratic formula to find the zeros of the function.
$x=\frac{-b\pm\sqrt{b^{2}-ac}}{2a}$

STEP 4

Substitute the values of $a$ , $b$ , and $c$ into the formula.
$x=\frac{-10\pm\sqrt{10^{2}-4(4)()}}{2(4)}$

STEP 5

implify the expression under the square root.
$x=\frac{-10\pm\sqrt{100-80}}{8}$

STEP 6

Further simplify the expression.
$x=\frac{-10\pm\sqrt{20}}{8}$

STEP 7

implify the square root.
$x=\frac{-10\pm2\sqrt{5}}{}$

STEP 8

Divide each term in the numerator by the denominator.
$x=-\frac{5}{4}\pm\frac{\sqrt{5}}{4}$ The zeros of the function are $-\frac{5}{4}+\frac{\sqrt{5}}{4}$ and $-\frac{5}{4}-\frac{\sqrt{5}}{4}$ .

STEP 9

The zeros of the function are also the x-intercepts of the graph of the function. Therefore, the x-intercepts of the graph of the function are $-\frac{5}{4}+\frac{\sqrt{5}}{4}$ and $-\frac{5}{4}-\frac{\sqrt{5}}{4}$ .
So, the correct choice is A. The zeros and the x-intercepts are the same. They are $-\frac{5}{4}+\frac{\sqrt{5}}{4}$ and $-\frac{5}{4}-\frac{\sqrt{5}}{4}$ .

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