QuestionFind the real zeros and x-intercepts of the function $f(x)=x^{2}+10x+22$ using the quadratic formula.

Studdy Solution

STEP 1

Assumptions1. The function given is a quadratic function, $f(x)=x^{}+10x+22$ . . We need to find the real zeros of the function, if any, using the quadratic formula.
3. 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 $ax^{}+bx+c=0$ .

STEP 2

Identify the coefficients $a$ , $b$ , and $c$ from the given quadratic function.
In the given function $f(x)=x^{2}+10x+22$ , we have $a=1$ , $b=10$ , and $c=22$ .

STEP 3

Substitute the coefficients $a$ , $b$ , and $c$ into the quadratic formula.
$x=\frac{-b\pm\sqrt{b^{2}-ac}}{2a}$ becomes $x=\frac{-10\pm\sqrt{10^{2}-(1)(22)}}{2(1)}$

STEP 4

implify the expression under the square root.
$x=\frac{-10\pm\sqrt{100-88}}{2}$

STEP 5

Calculate the value under the square root.
$x=\frac{-10\pm\sqrt{12}}{2}$

STEP 6

implify the square root.
$x=\frac{-10\pm2\sqrt{3}}{2}$

STEP 7

implify the expression to find the values of $x$ .
$x=-5\pm\sqrt{3}$ These are the real zeros of the function.

STEP 8

The x-intercepts of a function are the values of $x$ for which $f(x)=0$ . In other words, they are the real zeros of the function.
Since the real zeros of the function are $x=-5+\sqrt{3}$ and $x=-5-\sqrt{3}$ , these are also the x-intercepts of the function.
So, the answer is B. The zeros and the x-intercepts are the same. They are $x=-5+\sqrt{3}$ and $x=-5-\sqrt{3}$ .

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