To solve the quadratic equation \(-x^2 + x = -10\), we first rearrange it into standard form:
\[ -x^2 + x + 10 = 0 \]
Next, we multiply through by -1 to make the leading coefficient positive:
\[ x^2 - x - 10 = 0 \]
Now, we can use the quadratic formula, which is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -1\), and \(c = -10\). Plugging in these values gives:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (-1)^2 - 4 \cdot 1 \cdot (-10) = 1 + 40 = 41 \]
Now, substituting back into the formula:
\[ x = \frac{1 \pm \sqrt{41}}{2} \]
Next, we approximate \(\sqrt{41}\):
\[ \sqrt{41} \approx 6.4 \]
Now we can find the two possible values for \(x\):
\[ x_1 = \frac{1 + 6.4}{2} = \frac{7.4}{2} \approx 3.7 \]
\[ x_2 = \frac{1 - 6.4}{2} = \frac{-5.4}{2} \approx -2.7 \]
Rounding to the nearest hundredth, the solutions to the equation \(-x^2 + x = -10\) are approximately:
\[ x_1 \approx 3.70 \quad \text{and} \quad x_2 \approx -2.70 \]
So the solutions are \(3.70\) and \(-2.70\).