Question

-x^2 + x = -10 rounded to the nearest hundredth

Answer 1

To solve the equation \(-x^2 + x = -10\), we first rearrange it to a standard quadratic form. By adding 10 to both sides, we get:

\[
-x^2 + x + 10 = 0
\]

Next, we'll multiply through by -1 to make the leading coefficient positive:

\[
x^2 - x - 10 = 0
\]

Now we can use the quadratic formula to solve for \(x\):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 1\), \(b = -1\), and \(c = -10\). Plugging in the values:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1}
\]

Calculating the discriminant:

\[
b^2 - 4ac = 1 + 40 = 41
\]

Now substituting back into the formula:

\[
x = \frac{1 \pm \sqrt{41}}{2}
\]

Calculating \(\sqrt{41}\):

\[
\sqrt{41} \approx 6.4031
\]

Now substitute it back into the expression:

\[
x \approx \frac{1 \pm 6.4031}{2}
\]

Calculating the two possible values for \(x\):

1. \(x_1 = \frac{1 + 6.4031}{2} \approx \frac{7.4031}{2} \approx 3.70155\)
2. \(x_2 = \frac{1 - 6.4031}{2} \approx \frac{-5.4031}{2} \approx -2.70155\)

Rounding these results to the nearest hundredth, we get:

1. \(x_1 \approx 3.70\)
2. \(x_2 \approx -2.70\)

Thus, the solutions to the equation \(-x^2 + x = -10\) rounded to the nearest hundredth are:

\[
\boxed{3.70} \quad \text{and} \quad \boxed{-2.70}
\]