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 form:

\[
-2x^2 + 3x - 1 + 2 = 0
\]
\[
-2x^2 + 3x + 1 = 0
\]

Now, we can use the quadratic formula, which is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = -2\), \(b = 3\), and \(c = 1\). First, we calculate the discriminant (\(b^2 - 4ac\)):

\[
b^2 - 4ac = 3^2 - 4(-2)(1) = 9 + 8 = 17
\]

Now we plug the values into the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{17}}{2(-2)}
\]
\[
x = \frac{-3 \pm \sqrt{17}}{-4}
\]
\[
x = \frac{3 \mp \sqrt{17}}{4}
\]

This gives us two solutions:

1. \(x_1 = \frac{3 + \sqrt{17}}{4}\)
2. \(x_2 = \frac{3 - \sqrt{17}}{4}\)

Next, we calculate the approximate values of these solutions:

1. Calculate \(\sqrt{17} \approx 4.123\):
\[
x_1 \approx \frac{3 + 4.123}{4} = \frac{7.123}{4} \approx 1.78075 \approx 1.78
\]

2. Calculate:
\[
x_2 \approx \frac{3 - 4.123}{4} = \frac{-1.123}{4} \approx -0.28075 \approx -0.28
\]

The solutions rounded to the nearest hundredths are:

- Smaller solution: \(x \approx -0.28\)
- Larger solution: \(x \approx 1.78\)

Final answer:

\[
x = -0.28 \quad \text{or} \quad 1.78
\]