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Answer :
To find the zeros of the function [tex]\(y = 2x^2 + 9x + 4\)[/tex], we need to solve the equation [tex]\(2x^2 + 9x + 4 = 0\)[/tex].
We can use the quadratic formula to find the solutions. The quadratic formula is:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 9 \)[/tex]
- [tex]\( c = 4 \)[/tex]
Let's calculate each part:
1. Calculate the discriminant:
[tex]\[ b^2 - 4ac = 9^2 - 4 \times 2 \times 4 = 81 - 32 = 49 \][/tex]
2. Find the square root of the discriminant:
[tex]\[ \sqrt{49} = 7 \][/tex]
3. Apply the quadratic formula:
- First solution ([tex]\(x_1\)[/tex]):
[tex]\[ x_1 = \frac{{-9 + 7}}{2 \times 2} = \frac{{-2}}{4} = -0.5 \][/tex]
- Second solution ([tex]\(x_2\)[/tex]):
[tex]\[ x_2 = \frac{{-9 - 7}}{2 \times 2} = \frac{{-16}}{4} = -4 \][/tex]
Therefore, the zeros of the function are [tex]\(x = -0.5\)[/tex] and [tex]\(x = -4\)[/tex].
So the correct answer is:
D. [tex]\( x = -\frac{1}{2}, x = -4 \)[/tex]
We can use the quadratic formula to find the solutions. The quadratic formula is:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 9 \)[/tex]
- [tex]\( c = 4 \)[/tex]
Let's calculate each part:
1. Calculate the discriminant:
[tex]\[ b^2 - 4ac = 9^2 - 4 \times 2 \times 4 = 81 - 32 = 49 \][/tex]
2. Find the square root of the discriminant:
[tex]\[ \sqrt{49} = 7 \][/tex]
3. Apply the quadratic formula:
- First solution ([tex]\(x_1\)[/tex]):
[tex]\[ x_1 = \frac{{-9 + 7}}{2 \times 2} = \frac{{-2}}{4} = -0.5 \][/tex]
- Second solution ([tex]\(x_2\)[/tex]):
[tex]\[ x_2 = \frac{{-9 - 7}}{2 \times 2} = \frac{{-16}}{4} = -4 \][/tex]
Therefore, the zeros of the function are [tex]\(x = -0.5\)[/tex] and [tex]\(x = -4\)[/tex].
So the correct answer is:
D. [tex]\( x = -\frac{1}{2}, x = -4 \)[/tex]
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