To solve the equation \(-2x^2 + 3x - 1 = -2\) using the quadratic formula, we first need to rearrange the equation into standard form \(ax^2 + bx + c = 0\).
Starting with:
\[ -2x^2 + 3x - 1 + 2 = 0 \]
This simplifies to:
\[ -2x^2 + 3x + 1 = 0 \]
Now, we can identify the coefficients:
- \(a = -2\)
- \(b = 3\)
- \(c = 1\)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now, plug in the values of \(a\), \(b\), and \(c\):
- Calculate \(b^2 - 4ac\):
\[ b^2 - 4ac = 3^2 - 4(-2)(1) = 9 + 8 = 17 \]
- Use the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{17}}{2(-2)} = \frac{-3 \pm \sqrt{17}}{-4} \]
- This results in two potential solutions:
\[ x = \frac{-3 + \sqrt{17}}{-4} \quad \text{and} \quad x = \frac{-3 - \sqrt{17}}{-4} \]
Let’s calculate both:
- First solution:
\[ x_1 = \frac{-3 + \sqrt{17}}{-4} \] \[ \sqrt{17} \approx 4.1231 \] Thus,
\[ x_1 = \frac{-3 + 4.1231}{-4} = \frac{1.1231}{-4} \approx -0.2808 \]
- Second solution:
\[ x_2 = \frac{-3 - \sqrt{17}}{-4} \] Thus,
\[ x_2 = \frac{-3 - 4.1231}{-4} = \frac{-7.1231}{-4} \approx 1.7808 \]
Now we round both solutions to the nearest hundredths:
- \(x_1 \approx -0.28\)
- \(x_2 \approx 1.78\)
So, the final solutions, with the smaller solution first, are:
\[ x = -0.28 \quad \text{or} \quad x = 1.78 \]