Which of the following is a solution to the quadratic equation −x^2+x=−10

? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses

x=0.27
x equals 0.27

x=1.7
x equals 1.7

x=3.7
x equals 3.7

x=0.77
x equals 0.77

1 answer

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

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

Next, we can multiply the entire equation by \(-1\) to simplify it:

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

Now, we can use the quadratic formula to find the solutions, where \(a = 1\), \(b = -1\), and \(c = -10\):

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

Substituting in the values:

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

Calculating inside the square root:

\[ x = \frac{1 \pm \sqrt{1 + 40}}{2} \] \[ x = \frac{1 \pm \sqrt{41}}{2} \]

Calculating \(\sqrt{41}\):

\[ \sqrt{41} \approx 6.403 \]

Now substituting back to find \(x\):

\[ x = \frac{1 + 6.403}{2} \quad \text{and} \quad x = \frac{1 - 6.403}{2} \] \[ x \approx \frac{7.403}{2} \approx 3.7015 \quad \text{and} \quad x \approx \frac{-5.403}{2} \approx -2.7015 \]

Thus, rounding \(3.7015\) to the nearest hundredth gives \(3.70\).

So the solution to the quadratic equation \(-x^2+x=-10\) is approximately:

\[ x \approx 3.70 \]

Among the provided options, \(x = 3.7\) (or \(x\) equals \(3.7\)) is the correct answer.