Question

Use the quadratic formula to solve the equation.
negative 5−5x squaredx2negative 9−9xplus+22equals=0
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
xequals=enter your response here
​(Use a comma to separate answers as​ needed.)
B.
The solution is not a real number.

Answer 1

To solve the equation \(-5x^2 - 9x + 22 = 0\) using the quadratic formula, we will follow these steps:

1. Identify the coefficients \(a\), \(b\), and \(c\) from the equation in standard form \(ax^2 + bx + c = 0\):
\[
a = -5, \quad b = -9, \quad c = 22
\]

2. Use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

3. Calculate \(b^2 - 4ac\):
\[
b^2 = (-9)^2 = 81
\]
\[
4ac = 4 \times (-5) \times 22 = -440
\]
\[
b^2 - 4ac = 81 - (-440) = 81 + 440 = 521
\]

4. Since \(b^2 - 4ac = 521\) is greater than zero, there will be two real solutions.

5. Substitute back into the quadratic formula:
\[
x = \frac{-(-9) \pm \sqrt{521}}{2(-5)} = \frac{9 \pm \sqrt{521}}{-10}
\]

6. This gives us two solutions:
\[
x_1 = \frac{9 + \sqrt{521}}{-10}, \quad x_2 = \frac{9 - \sqrt{521}}{-10}
\]

7. To simplify, these can be left in the above form, or you can calculate approximate numerical values:
- Calculate \( \sqrt{521} \) using a calculator.
- Let's approximate \( \sqrt{521} \approx 22.845\).

Thus approximations for the solutions:
\[
x_1 \approx \frac{9 + 22.845}{-10} \approx \frac{31.845}{-10} \approx -3.1845
\]
\[
x_2 \approx \frac{9 - 22.845}{-10} \approx \frac{-13.845}{-10} \approx 1.3845
\]

Putting it in the final answer:
\[
\text{A. } x = \frac{9 + \sqrt{521}}{-10}, \, \frac{9 - \sqrt{521}}{-10} \quad \text{(or approximately } -3.1845, 1.3845 \text{)}
\]

Choose the appropriate response based on whether you need exact roots (with radical) or approximate decimal answers.

If using the radical form:
A. \( x = \frac{9 + \sqrt{521}}{-10}, \frac{9 - \sqrt{521}}{-10} \)

If using decimal approximation:
A. \( x \approx -3.1845, 1.3845 \)