To find the solutions to the quadratic equation \(-x^2 + x = -10\), we first rearrange it into standard form.
Starting with:
\[ -x^2 + x + 10 = 0 \]
We can multiply the entire equation by -1 to make the leading coefficient positive:
\[ x^2 - x - 10 = 0 \]
Now we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -1\), and \(c = -10\).
Calculating the discriminant:
\[ b^2 - 4ac = (-1)^2 - 4(1)(-10) = 1 + 40 = 41 \]
Now substituting back into the quadratic formula:
\[ x = \frac{-(-1) \pm \sqrt{41}}{2(1)} = \frac{1 \pm \sqrt{41}}{2} \]
Calculating \(\sqrt{41} \approx 6.4\), we find:
\[ x \approx \frac{1 + 6.4}{2} \approx \frac{7.4}{2} \approx 3.7 \] and \[ x \approx \frac{1 - 6.4}{2} \approx \frac{-5.4}{2} \approx -2.7 \]
So the solutions to the equation are approximately \(3.7\) and \(-2.7\).
From the given options, the correct solution rounded to the nearest hundredth is:
x equals 3.7.