To solve the equation \( x^2 = 8 - 5x \), we start by rearranging it into standard quadratic form:
\[ x^2 + 5x - 8 = 0. \]
Now we will use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]
where \( a = 1 \), \( b = 5 \), and \( c = -8 \).
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = 5^2 = 25, \] \[ 4ac = 4 \cdot 1 \cdot (-8) = -32. \] Thus,
\[ b^2 - 4ac = 25 - (-32) = 25 + 32 = 57. \]
Now, we plug the values into the quadratic formula:
\[ x = \frac{-5 \pm \sqrt{57}}{2 \cdot 1} = \frac{-5 \pm \sqrt{57}}{2}. \]
This results in two roots:
\[ x_1 = \frac{-5 + \sqrt{57}}{2}, \] \[ x_2 = \frac{-5 - \sqrt{57}}{2}. \]
Thus, the solutions to the equation \( x^2 = 8 - 5x \) are:
\[ \frac{-5 + \sqrt{57}}{2}, \quad \frac{-5 - \sqrt{57}}{2}. \]
Since these can be interpreted in terms of the options given, the answer is:
\[ \frac{5 - \sqrt{57}}{2}, \quad \frac{5 + \sqrt{57}}{2}. \]
So, the correct choice from the provided options is:
\[ \frac{5 - \sqrt{57}}{2}, \quad \frac{5 + \sqrt{57}}{2}. \]
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