Question

Find the solution to the quadratic equation - x ^ 2 + x = - 10 Assume that the solution has been rounded to the nearest hundredth, if applicable.
x = 0.77
x = 1.7
x = 3.7
x = 0.27

Answer 1

To solve the quadratic equation \(-x^2 + x = -10\), we first need to rearrange the equation into standard form.

1. Add \(10\) to both sides to get:
\[
-x^2 + x + 10 = 0
\]

2. Multiply the entire equation by \(-1\) to make the coefficient of \(x^2\) positive:
\[
x^2 - x - 10 = 0
\]

Now we can apply the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = -1\), and \(c = -10\).

3. Calculate the discriminant (\(b^2 - 4ac\)):
\[
(-1)^2 - 4 \times 1 \times (-10) = 1 + 40 = 41
\]

4. Plug the values into the quadratic formula:
\[
x = \frac{-(-1) \pm \sqrt{41}}{2 \times 1} = \frac{1 \pm \sqrt{41}}{2}
\]

5. Calculate \( \sqrt{41} \):
\[
\sqrt{41} \approx 6.403
\]

6. Substitute back to find the two possible solutions:
\[
x = \frac{1 + 6.403}{2} \approx \frac{7.403}{2} \approx 3.7015
\]
\[
x = \frac{1 - 6.403}{2} \approx \frac{-5.403}{2} \approx -2.7015
\]

Now rounding the first solution \(3.7015\) to the nearest hundredth gives us \(3.70\).

Thus, among the provided options of \(0.77\), \(1.7\), \(3.7\), and \(0.27\), the closest rounded solution is:
\[
\boxed{3.7}
\]