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Solve −2x^2 +3x−1=−2

using the quadratic formula. Round the solution(s) to the nearest hundredths. Enter the smaller solution first.(1 point)
x=___
or
___

1 answer

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

Starting with the original equation:

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

Add 2 to both sides to get:

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

This simplifies to:

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

Now, we can use the quadratic formula to find the values of \(x\). The quadratic formula is:

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

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

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

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

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

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

This simplifies to:

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

Which can be written as:

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

Now, we calculate the two potential solutions:

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

Calculating \( \sqrt{17} \):

\[ \sqrt{17} \approx 4.123 \]

Now substituting back into the equations:

  1. \(x_1 = \frac{3 + 4.123}{4} = \frac{7.123}{4} \approx 1.78075\)
  2. \(x_2 = \frac{3 - 4.123}{4} = \frac{-1.123}{4} \approx -0.28075\)

Rounding these solutions to the nearest hundredths:

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

So, the final rounded solutions are:

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

Thus, the complete answer is:

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