To solve the quadratic equation [tex]\(x^2 + 9x + 10 = 0\)[/tex] using the quadratic formula, we'll follow these steps:
1. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]. Here:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 9\)[/tex]
- [tex]\(c = 10\)[/tex]
2. Write down the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant:
[tex]\[
b^2 - 4ac = 9^2 - 4 \cdot 1 \cdot 10 = 81 - 40 = 41
\][/tex]
4. Since the discriminant is positive, there are two real solutions. Substitute the discriminant and the coefficients into the quadratic formula to find the solutions:
[tex]\[
x = \frac{-9 \pm \sqrt{41}}{2 \cdot 1}
\][/tex]
5. Calculate the two solutions separately:
First solution using the positive square root term:
[tex]\[
x_1 = \frac{-9 + \sqrt{41}}{2} \approx -1.2984378812835757
\][/tex]
Second solution using the negative square root term:
[tex]\[
x_2 = \frac{-9 - \sqrt{41}}{2} \approx -7.701562118716424
\][/tex]
6. Round the solutions to the nearest tenth:
[tex]\[
x_1 \approx -1.3
\][/tex]
[tex]\[
x_2 \approx -7.7
\][/tex]
So, the solutions to the equation [tex]\(x^2 + 9x + 10 = 0\)[/tex] are approximately [tex]\( x = -1.3 \)[/tex] and [tex]\( x = -7.7 \)[/tex].
Therefore, the correct answer is:
[tex]\[
x = -7.7 \text{ and } x = -1.3
\][/tex]
