Question

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

Answer 1

To solve the equation \(-2x^2 + 3x - 1 = -2\), we first need to rewrite it in standard form by adding 2 to both sides:

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

Now, we have the equation in the standard form \(ax^2 + bx + c = 0\), where \(a = -2\), \(b = 3\), and \(c = 1\).

We will use the quadratic formula to solve for \(x\):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

First, we calculate the discriminant \(b^2 - 4ac\):
\[
b^2 = 3^2 = 9
\]
\[
4ac = 4 \cdot (-2) \cdot 1 = -8
\]
\[
b^2 - 4ac = 9 - (-8) = 9 + 8 = 17
\]

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

This results in two solutions:
\[
x_1 = \frac{-3 + \sqrt{17}}{-4}
\]
\[
x_2 = \frac{-3 - \sqrt{17}}{-4}
\]

Calculating \( \sqrt{17} \):
\[
\sqrt{17} \approx 4.1231
\]

Now we can calculate the two expressions:
1. For \(x_1\):
\[
x_1 = \frac{-3 + 4.1231}{-4} \approx \frac{1.1231}{-4} \approx -0.2808
\]

2. For \(x_2\):
\[
x_2 = \frac{-3 - 4.1231}{-4} \approx \frac{-7.1231}{-4} \approx 1.7808
\]

Rounding both solutions to the nearest hundredths:
- \( x_1 \approx -0.28 \)
- \( x_2 \approx 1.78 \)

Thus, the smaller solution first is:
\[
\boxed{-0.28} \text{ and } \boxed{1.78}
\]