Answer :
To find the zeros of the function [tex]\( y = 2x^2 + 9x + 4 \)[/tex], we can use the quadratic formula, which is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are as follows:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 9 \)[/tex]
- [tex]\( c = 4 \)[/tex]
Step 1: Calculate the discriminant
The discriminant is the part of the quadratic formula under the square root, [tex]\( b^2 - 4ac \)[/tex].
[tex]\[
b^2 = 9^2 = 81
\][/tex]
[tex]\[
4ac = 4 \times 2 \times 4 = 32
\][/tex]
[tex]\[
\text{discriminant} = b^2 - 4ac = 81 - 32 = 49
\][/tex]
Step 2: Apply the quadratic formula
Now we can find the zero values of [tex]\( x \)[/tex] using the quadratic formula with the calculated discriminant.
Calculate [tex]\( x_1 \)[/tex]:
[tex]\[
x_1 = \frac{-b + \sqrt{\text{discriminant}}}{2a} = \frac{-9 + \sqrt{49}}{4} = \frac{-9 + 7}{4} = \frac{-2}{4} = -0.5
\][/tex]
Calculate [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = \frac{-b - \sqrt{\text{discriminant}}}{2a} = \frac{-9 - \sqrt{49}}{4} = \frac{-9 - 7}{4} = \frac{-16}{4} = -4.0
\][/tex]
Therefore, the zeros of the function [tex]\( y = 2x^2 + 9x + 4 \)[/tex] are:
[tex]\( x = -0.5 \)[/tex] and [tex]\( x = -4.0 \)[/tex]
This corresponds to answer choice D: [tex]\( x = -\frac{1}{2}, x = -4 \)[/tex].
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are as follows:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 9 \)[/tex]
- [tex]\( c = 4 \)[/tex]
Step 1: Calculate the discriminant
The discriminant is the part of the quadratic formula under the square root, [tex]\( b^2 - 4ac \)[/tex].
[tex]\[
b^2 = 9^2 = 81
\][/tex]
[tex]\[
4ac = 4 \times 2 \times 4 = 32
\][/tex]
[tex]\[
\text{discriminant} = b^2 - 4ac = 81 - 32 = 49
\][/tex]
Step 2: Apply the quadratic formula
Now we can find the zero values of [tex]\( x \)[/tex] using the quadratic formula with the calculated discriminant.
Calculate [tex]\( x_1 \)[/tex]:
[tex]\[
x_1 = \frac{-b + \sqrt{\text{discriminant}}}{2a} = \frac{-9 + \sqrt{49}}{4} = \frac{-9 + 7}{4} = \frac{-2}{4} = -0.5
\][/tex]
Calculate [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = \frac{-b - \sqrt{\text{discriminant}}}{2a} = \frac{-9 - \sqrt{49}}{4} = \frac{-9 - 7}{4} = \frac{-16}{4} = -4.0
\][/tex]
Therefore, the zeros of the function [tex]\( y = 2x^2 + 9x + 4 \)[/tex] are:
[tex]\( x = -0.5 \)[/tex] and [tex]\( x = -4.0 \)[/tex]
This corresponds to answer choice D: [tex]\( x = -\frac{1}{2}, x = -4 \)[/tex].
