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 \(-2x^2 + 3x - 1 = -2\), we first rewrite it in standard form. We do this by adding 2 to both sides:

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

This simplifies to:

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

Next, we multiply the whole equation by -1 to make the coefficient of \(x^2\) positive:

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

Now, we can use the quadratic formula, which is given by:

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

where \(a = 2\), \(b = -3\), and \(c = -1\).

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

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

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

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

Now we compute the two potential solutions:

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

Now, we can approximate these values. The square root of 17 is approximately 4.1231.

Calculating \(x_1\):

\[
x_1 = \frac{3 + 4.1231}{4} = \frac{7.1231}{4} \approx 1.7808
\]

Calculating \(x_2\):

\[
x_2 = \frac{3 - 4.1231}{4} = \frac{-1.1231}{4} \approx -0.2808
\]

Finally, rounding to the nearest hundredths:

- Smaller solution \(x_2 \approx -0.28\)
- Larger solution \(x_1 \approx 1.78\)

Thus, the solutions rounded to the nearest hundredths are:

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