Question

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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 \(-2x^2 + 3x - 1 = -2\) using the quadratic formula, we first need to rewrite the equation in standard form \(ax^2 + bx + c = 0\).

Starting with the original equation:

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

Simplifying this gives:

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

Now, we identify the coefficients as follows:

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

The quadratic formula is given by:

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

Now, let's compute the discriminant (\(b^2 - 4ac\)):

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

Now, substituting into the quadratic formula:

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

Now, we need to evaluate the two solutions:

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

Calculating \(\sqrt{17}\):

\(\sqrt{17} \approx 4.123\)

Now calculating the solutions:

1. \(x_1 = \frac{3 + 4.123}{4} \approx \frac{7.123}{4} \approx 1.78075\)
2. \(x_2 = \frac{3 - 4.123}{4} \approx \frac{-1.123}{4} \approx -0.28075\)

Rounding both solutions to the nearest hundredths gives:

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

Thus, the smaller solution first, followed by the larger solution, is:

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