To solve the equation \(-2x^2 + 3x - 1 = -2\), we first rearrange it into standard quadratic form:
\[ -2x^2 + 3x - 1 + 2 = 0 \] \[ -2x^2 + 3x + 1 = 0 \]
Now we will apply the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, the coefficients are:
- \(a = -2\)
- \(b = 3\)
- \(c = 1\)
First, we compute the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = 3^2 - 4(-2)(1) = 9 + 8 = 17 \]
Now we substitute the values into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{17}}{2(-2)} \] \[ x = \frac{-3 \pm \sqrt{17}}{-4} \]
This simplifies to:
\[ x = \frac{3 \mp \sqrt{17}}{4} \]
Now we calculate the two possible solutions:
- For \(x_1 = \frac{3 + \sqrt{17}}{4}\)
- For \(x_2 = \frac{3 - \sqrt{17}}{4}\)
Calculating \(\sqrt{17}\):
\[ \sqrt{17} \approx 4.1231 \]
Now substituting back to find \(x_1\) and \(x_2\):
- Calculate \(x_1\):
\[ x_1 = \frac{3 + 4.1231}{4} = \frac{7.1231}{4} \approx 1.7808 \]
- Calculate \(x_2\):
\[ x_2 = \frac{3 - 4.1231}{4} = \frac{-1.1231}{4} \approx -0.2808 \]
Finally, rounding these solutions to the nearest hundredths:
- The smaller solution \(x_2 \approx -0.28\)
- The larger solution \(x_1 \approx 1.78\)
Thus, the two solutions rounded to the nearest hundredths are:
\[ \boxed{-0.28} \quad \text{and} \quad \boxed{1.78} \]
1 answer