To solve the quadratic equation \(-x^2 + x = -10\), we can rearrange it into standard form:
\[ -x^2 + x + 10 = 0 \]
Multiplying through by -1 to make the leading coefficient positive gives us:
\[ x^2 - x - 10 = 0 \]
Now we can use the quadratic formula, which is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = -1\), and \(c = -10\).
Calculating the discriminant:
\[ b^2 - 4ac = (-1)^2 - 4(1)(-10) = 1 + 40 = 41 \]
Now substitute back into the quadratic formula:
\[ x = \frac{-(-1) \pm \sqrt{41}}{2(1)} = \frac{1 \pm \sqrt{41}}{2} \]
Calculating \(\sqrt{41}\), which is approximately 6.403:
\[ x = \frac{1 \pm 6.403}{2} \]
This gives us two potential solutions:
- \(x = \frac{1 + 6.403}{2} = \frac{7.403}{2} \approx 3.7015\)
- \(x = \frac{1 - 6.403}{2} = \frac{-5.403}{2} \approx -2.7015\)
Rounding to the nearest hundredth,
- \(x \approx 3.70\)
- \(x \approx -2.70\)
Out of the provided options, the closest solution is:
x = 3.7.