To solve the quadratic equation [tex]\(2x^2 + 11x + 9 = 0\)[/tex] using the quadratic formula, follow these steps:
1. Identify coefficients:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 11\)[/tex]
- [tex]\(c = 9\)[/tex]
2. Write down the quadratic formula: The quadratic formula for solving [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Plug the values [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
\text{Discriminant} = 11^2 - 4 \cdot 2 \cdot 9 = 121 - 72 = 49
\][/tex]
4. Find the square root of the discriminant:
[tex]\[
\sqrt{49} = 7
\][/tex]
5. Calculate the two solutions using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a}
\][/tex]
Substitute [tex]\(b = 11\)[/tex], [tex]\(\sqrt{\text{Discriminant}} = 7\)[/tex], and [tex]\(a = 2\)[/tex]:
[tex]\[
x_1 = \frac{-11 + 7}{2 \cdot 2} = \frac{-11 + 7}{4} = \frac{-4}{4} = -1
\][/tex]
[tex]\[
x_2 = \frac{-11 - 7}{2 \cdot 2} = \frac{-11 - 7}{4} = \frac{-18}{4} = -4.5
\][/tex]
So, the solutions to the equation [tex]\(2x^2 + 11x + 9 = 0\)[/tex] are [tex]\(x = -1\)[/tex] and [tex]\(x = -4.5\)[/tex].
Based on these solutions, the correct answer is:
B. [tex]\(x = -1, x = -9/2\)[/tex]
