QuestionFind the zeros of $f(x)=2x^2+10x+9$ and determine their relationship to the $x$ -intercepts.

Studdy Solution

STEP 1

Assumptions1. The function is given by $f(x)=x^{}+10x+9$ . We are looking for the zeros of the function, which are also the x-intercepts of the graph of the function.

STEP 2

The zeros of the function are the values of $x$ that make the function equal to zero. We can find these by setting the function equal to zero and solving for $x$ .
$0 =2x^{2} +10x +9$

STEP 3

This is a quadratic equation of the form $ax^{2} + bx + c =0$ , and it can be solved using the quadratic formula, which is given by $x = \frac{-b \pm \sqrt{b^{2} -ac}}{2a}$

STEP 4

In our equation, $a =2$ , $b =10$ , and $c =9$ . Substituting these values into the quadratic formula gives $x = \frac{-10 \pm \sqrt{10^{2} -4 \cdot2 \cdot9}}{2 \cdot2}$

STEP 5

implify the expression under the square root $x = \frac{-10 \pm \sqrt{100 -72}}{4}$

STEP 6

Further simplify the expression $x = \frac{-10 \pm \sqrt{28}}{4}$

STEP 7

The square root of28 can be simplified to $2\sqrt{7}$ , so the expression becomes $x = \frac{-10 \pm2\sqrt{7}}{4}$

STEP 8

We can simplify the expression further by dividing each term in the numerator by2 $x = \frac{-5 \pm \sqrt{7}}{2}$ So, the zeros of the function are $x = -5 + \sqrt{7}/2$ and $x = -5 - \sqrt{7}/2$ .

STEP 9

The zeros of the function are also the x-intercepts of the graph of the function. Therefore, the x-intercepts are $x = -5 + \sqrt{7}/2$ and $x = -5 - \sqrt{7}/2$ .
So, the correct choice is A. The zeros and the x-intercepts are the same. They are $x = -5 + \sqrt{7}/2$ and $x = -5 - \sqrt{7}/2$ .

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