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 \(-2x^2 + 3x - 1 = -2\), we first simplify it by moving all terms to one side:

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

This simplifies to:

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

For the quadratic formula, we generally use the form \(ax^2 + bx + c = 0\). Here, we have:

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

The quadratic formula is given by:

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

First, we calculate the discriminant \(\Delta = b^2 - 4ac\):

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

Now we can use the quadratic formula to find the solutions:

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

This simplifies to:

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

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

1. Using the plus sign:

\[
x_1 = \frac{{3 + \sqrt{17}}}{4}
\]

2. Using the minus sign:

\[
x_2 = \frac{{3 - \sqrt{17}}}{4}
\]

Next, we will calculate \(\sqrt{17}\):

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

Now substituting this value in:

1. For \(x_1\):

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

Rounding to the nearest hundredths gives:

\[
x_1 \approx 1.78
\]

2. For \(x_2\):

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

Rounding to the nearest hundredths gives:

\[
x_2 \approx -0.28
\]

Thus, the final solutions, with the smaller solution first, are:

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