Question

Solve −2x2+3x−1=−2
using the quadratic formula. Round the solution(s) to the nearest hundredths. Enter the smaller solution first.(1 point)
x=
or

Answer 1

To solve the equation \(-2x^2 + 3x - 1 = -2\), we first rearrange it to standard form:

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

This simplifies to:

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

Next, we'll use the quadratic formula, which states:

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

In our equation, the coefficients are:

- \(a = -2\)
- \(b = 3\)
- \(c = 1\)

Calculating the discriminant \(b^2 - 4ac\):

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

Now we can substitute back into the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{17}}{2 \times -2} = \frac{-3 \pm \sqrt{17}}{-4}
\]

This simplifies to:

\[
x = \frac{3 \mp \sqrt{17}}{4}
\]

Now we calculate the two possible values of \(x\):

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

Calculating \(x_1\) and \(x_2\):

First, we find \(\sqrt{17} \approx 4.123\).

Calculating \(x_1\):

\[
x_1 = \frac{3 + 4.123}{4} = \frac{7.123}{4} \approx 1.78075
\]

Calculating \(x_2\):

\[
x_2 = \frac{3 - 4.123}{4} = \frac{-1.123}{4} \approx -0.28075
\]

Rounding to the nearest hundredths:

1. \(x_1 \approx 1.78\)
2. \(x_2 \approx -0.28\)

Thus, the solutions to the equation are:

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

So, we write the final answer as:

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