Question

Solve −2x^2+3x−1=−2
using the quadratic formula. Round the solution(s) to the nearest hundredths. Enter the smaller solution first.

x= __ or __

Answer 1

To solve the equation \(-2x^2 + 3x - 1 = -2\), we first rearrange it into standard quadratic form \(ax^2 + bx + c = 0\).

1. **Add 2 to both sides:**
\[
-2x^2 + 3x - 1 + 2 = 0
\]
This simplifies to:
\[
-2x^2 + 3x + 1 = 0
\]

2. **Now, let's identify the coefficients \(a\), \(b\), and \(c\):**
- \(a = -2\)
- \(b = 3\)
- \(c = 1\)

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

4. **Calculate the discriminant \(b^2 - 4ac\):**
\[
b^2 - 4ac = 3^2 - 4(-2)(1) = 9 + 8 = 17
\]

5. **Now apply the values into the quadratic formula:**
\[
x = \frac{-3 \pm \sqrt{17}}{2(-2)} = \frac{-3 \pm \sqrt{17}}{-4}
\]

6. **Splitting into two solutions:**
\[
x_1 = \frac{-3 + \sqrt{17}}{-4}, \quad x_2 = \frac{-3 - \sqrt{17}}{-4}
\]

7. **Calculating \(\sqrt{17}\):**
- \(\sqrt{17} \approx 4.1231\)

8. **Finding \(x_1\) and \(x_2\):**
\[
x_1 = \frac{-3 + 4.1231}{-4} = \frac{1.1231}{-4} \approx -0.2808 \implies -0.28 \text{ (rounded to hundredths)}
\]
\[
x_2 = \frac{-3 - 4.1231}{-4} = \frac{-7.1231}{-4} \approx 1.7808 \implies 1.78 \text{ (rounded to hundredths)}
\]

9. **Final solutions:**
The two solutions are:
\[
x \approx -0.28 \quad \text{or} \quad x \approx 1.78
\]

Thus, the final answer is:
\[
x = -0.28 \quad \text{or} \quad 1.78
\]