Question

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

Answer 1

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

Start by adding 2 to both sides:

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

This simplifies to:

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

Now, we can identify \(a = -2\), \(b = 3\), and \(c = 1\).

The quadratic formula is given by:

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

Now, we calculate the discriminant:

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

Now we can substitute into the quadratic formula:

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

This gives us two solutions:

\[
x = \frac{-3 + \sqrt{17}}{-4} \quad \text{and} \quad x = \frac{-3 - \sqrt{17}}{-4}
\]

Calculating each solution:

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

Now we'll compute the numerical values:

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

1. For \(x_1\):

\[
x_1 = \frac{-3 + 4.123}{-4} = \frac{1.123}{-4} \approx -0.28075 \quad \text{(rounded to -0.28)}
\]

2. For \(x_2\):

\[
x_2 = \frac{-3 - 4.123}{-4} = \frac{-7.123}{-4} \approx 1.78075 \quad \text{(rounded to 1.78)}
\]

Thus, the two solutions are approximately \(x_1 \approx -0.28\) and \(x_2 \approx 1.78\).

Thus, rounded to the nearest hundredths, the final solutions are:

\[
\boxed{-0.28} \quad \text{and} \quad \boxed{1.78}
\]