To find the solutions of the quadratic equation \( x^2 = 7x + 4 \), we first rearrange it into standard form:
\[ x^2 - 7x - 4 = 0 \]
Now, we can use the quadratic formula to solve for \( x \), which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \( a = 1 \), \( b = -7 \), and \( c = -4 \). Plugging in these values:
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Calculate the discriminant: \[ b^2 - 4ac = (-7)^2 - 4(1)(-4) = 49 + 16 = 65 \]
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Plug the values into the quadratic formula: \[ x = \frac{-(-7) \pm \sqrt{65}}{2(1)} = \frac{7 \pm \sqrt{65}}{2} \]
Thus, the solutions are:
\[ x = \frac{7 - \sqrt{65}}{2} \quad \text{and} \quad x = \frac{7 + \sqrt{65}}{2} \]
This corresponds to the option:
\[ \frac{7 - \sqrt{65}}{2}, \frac{7 + \sqrt{65}}{2} \]
So the final answer is:
\(\frac{7 - \sqrt{65}}{2}, \frac{7 + \sqrt{65}}{2}\)