Answer :
To find the zeros of the function [tex]\( y = 2x^2 + 9x + 4 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] that make [tex]\( y = 0 \)[/tex]. These values, also known as roots, can be found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the quadratic equation [tex]\( 2x^2 + 9x + 4 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 9 \)[/tex]
- [tex]\( c = 4 \)[/tex]
Now let's follow the steps to find the roots:
1. Calculate the Discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac = 9^2 - 4 \times 2 \times 4 = 81 - 32 = 49
\][/tex]
2. Calculate the Square Root of the Discriminant:
[tex]\[
\sqrt{49} = 7
\][/tex]
3. Use the Quadratic Formula to Find the Roots:
- The first root:
[tex]\[
x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} = \frac{-9 + 7}{4} = \frac{-2}{4} = -0.5
\][/tex]
- The second root:
[tex]\[
x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{-9 - 7}{4} = \frac{-16}{4} = -4
\][/tex]
The zeros of the function [tex]\( y = 2x^2 + 9x + 4 \)[/tex] are [tex]\( x = -0.5 \)[/tex] and [tex]\( x = -4 \)[/tex]. Therefore, the answer is option C: [tex]\( x = -\frac{1}{2}, x = -4 \)[/tex].
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the quadratic equation [tex]\( 2x^2 + 9x + 4 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 9 \)[/tex]
- [tex]\( c = 4 \)[/tex]
Now let's follow the steps to find the roots:
1. Calculate the Discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac = 9^2 - 4 \times 2 \times 4 = 81 - 32 = 49
\][/tex]
2. Calculate the Square Root of the Discriminant:
[tex]\[
\sqrt{49} = 7
\][/tex]
3. Use the Quadratic Formula to Find the Roots:
- The first root:
[tex]\[
x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} = \frac{-9 + 7}{4} = \frac{-2}{4} = -0.5
\][/tex]
- The second root:
[tex]\[
x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{-9 - 7}{4} = \frac{-16}{4} = -4
\][/tex]
The zeros of the function [tex]\( y = 2x^2 + 9x + 4 \)[/tex] are [tex]\( x = -0.5 \)[/tex] and [tex]\( x = -4 \)[/tex]. Therefore, the answer is option C: [tex]\( x = -\frac{1}{2}, x = -4 \)[/tex].
