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 \(-2x^2 + 3x - 1 = -2\) using the quadratic formula, we first move all terms to one side of the equation:

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

This simplifies to:

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

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

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

In this case, \(a = -2\), \(b = 3\), and \(c = 1\).

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

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

Next, we substitute values into the quadratic formula:

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

This simplifies to:

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

Now, we calculate the two possible solutions:

1. Using the positive case:

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

2. Using the negative case:

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

Let's calculate \(\sqrt{17}\) (approximately 4.1231):

1. For \(x_1\):

\[ x_1 = \frac{-3 + 4.1231}{-4} = \frac{1.1231}{-4} \approx -0.2808 \]

2. For \(x_2\):

\[ x_2 = \frac{-3 - 4.1231}{-4} = \frac{-7.1231}{-4} \approx 1.7808 \]

Rounding these to the nearest hundredths:

1. \(x_1 \approx -0.28\)
2. \(x_2 \approx 1.78\)

So the final solutions, with the smaller one first, are:

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

Thus, the complete answer is:

\(x = -0.28 \text{ or } x = 1.78\)