Answer :
To find the zeros of the function [tex]\(y = 2x^2 + 9x + 4\)[/tex], we can use the quadratic formula. The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the quadratic equation [tex]\(2x^2 + 9x + 4\)[/tex], the coefficients are:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 9\)[/tex]
- [tex]\(c = 4\)[/tex]
1. Calculate the Discriminant:
The discriminant is given by the expression [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
b^2 = 9^2 = 81
\][/tex]
[tex]\[
4ac = 4 \times 2 \times 4 = 32
\][/tex]
[tex]\[
\text{Discriminant} = 81 - 32 = 49
\][/tex]
2. Calculate the Two Solutions:
Use the quadratic formula with the discriminant calculated above.
[tex]\[
x = \frac{-9 \pm \sqrt{49}}{2 \times 2}
\][/tex]
Calculate the square root of the discriminant:
[tex]\[
\sqrt{49} = 7
\][/tex]
First Solution (Positive Root):
[tex]\[
x_1 = \frac{-9 + 7}{4} = \frac{-2}{4} = -\frac{1}{2}
\][/tex]
Second Solution (Negative Root):
[tex]\[
x_2 = \frac{-9 - 7}{4} = \frac{-16}{4} = -4
\][/tex]
Thus, the zeros of the function are [tex]\(x = -\frac{1}{2}\)[/tex] and [tex]\(x = -4\)[/tex].
The correct answer is [tex]\(\text{B. } x = -\frac{1}{2}, x = -4\)[/tex].
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the quadratic equation [tex]\(2x^2 + 9x + 4\)[/tex], the coefficients are:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 9\)[/tex]
- [tex]\(c = 4\)[/tex]
1. Calculate the Discriminant:
The discriminant is given by the expression [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
b^2 = 9^2 = 81
\][/tex]
[tex]\[
4ac = 4 \times 2 \times 4 = 32
\][/tex]
[tex]\[
\text{Discriminant} = 81 - 32 = 49
\][/tex]
2. Calculate the Two Solutions:
Use the quadratic formula with the discriminant calculated above.
[tex]\[
x = \frac{-9 \pm \sqrt{49}}{2 \times 2}
\][/tex]
Calculate the square root of the discriminant:
[tex]\[
\sqrt{49} = 7
\][/tex]
First Solution (Positive Root):
[tex]\[
x_1 = \frac{-9 + 7}{4} = \frac{-2}{4} = -\frac{1}{2}
\][/tex]
Second Solution (Negative Root):
[tex]\[
x_2 = \frac{-9 - 7}{4} = \frac{-16}{4} = -4
\][/tex]
Thus, the zeros of the function are [tex]\(x = -\frac{1}{2}\)[/tex] and [tex]\(x = -4\)[/tex].
The correct answer is [tex]\(\text{B. } x = -\frac{1}{2}, x = -4\)[/tex].
