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 to standard form:

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

This simplifies to:

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

Next, we will apply the quadratic formula:

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

In our equation, \(a = -2\), \(b = 3\), and \(c = 1\).

Now, we calculate the discriminant \(b^2 - 4ac\):

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

Now, we can substitute \(a\), \(b\), and the discriminant into the quadratic formula:

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

This can be simplified to:

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

Now we calculate both solutions:

1. For the positive root:
\[
x_1 = \frac{3 + \sqrt{17}}{4}
\]

Approximating \(\sqrt{17} \approx 4.123\):
\[
x_1 \approx \frac{3 + 4.123}{4} \approx \frac{7.123}{4} \approx 1.78075
\]
Rounding to the nearest hundredth, \(x_1 \approx 1.78\).

2. For the negative root:
\[
x_2 = \frac{3 - \sqrt{17}}{4}
\]

Using \(\sqrt{17} \approx 4.123\):
\[
x_2 \approx \frac{3 - 4.123}{4} \approx \frac{-1.123}{4} \approx -0.28075
\]
Rounding to the nearest hundredth, \(x_2 \approx -0.28\).

So, the final answers rounded to the nearest hundredths are:

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

Thus, the final solutions:

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