To find the solutions to the quadratic equation \( -x^2 + x = -10 \), we first rearrange it into standard form:
\[ -x^2 + x + 10 = 0 \]
Multiplying through by -1 to make \( x^2 \) positive, we get:
\[ x^2 - x - 10 = 0 \]
Now, we can use the quadratic formula to find the solutions, where the quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \( x^2 - x - 10 = 0 \), we have:
- \( a = 1 \)
- \( b = -1 \)
- \( c = -10 \)
Now, we can calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-1)^2 - 4(1)(-10) = 1 + 40 = 41 \]
Next, we apply the quadratic formula:
\[ x = \frac{-(-1) \pm \sqrt{41}}{2(1)} = \frac{1 \pm \sqrt{41}}{2} \]
Calculating the two possible values for \( x \):
- \( x_1 = \frac{1 + \sqrt{41}}{2} \)
- \( x_2 = \frac{1 - \sqrt{41}}{2} \)
Calculating \( \sqrt{41} \) (approximately 6.4):
- \( x_1 \approx \frac{1 + 6.4}{2} = \frac{7.4}{2} = 3.7 \)
- \( x_2 \approx \frac{1 - 6.4}{2} = \frac{-5.4}{2} \approx -2.7 \)
Now, we round \( x_1 \) to the nearest hundredth and we get:
\[ x_1 \approx 3.7 \]
Thus, the solution to the quadratic equation \( -x^2 + x = -10 \) rounded to the nearest hundredth is:
\[ \boxed{3.7} \]