Sure! To solve the quadratic equation:
[tex]\[ x^2 = 7x + 4, \][/tex]
we start by rearranging it into the standard quadratic form \( ax^2 + bx + c = 0 \):
[tex]\[ x^2 - 7x - 4 = 0. \][/tex]
Here, the coefficients are:
- \( a = 1 \)
- \( b = -7 \)
- \( c = -4 \)
Next, we use the quadratic formula to solve for \( x \):
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \][/tex]
First, we calculate the discriminant \( \Delta \):
[tex]\[ \Delta = b^2 - 4ac = (-7)^2 - 4(1)(-4) = 49 + 16 = 65. \][/tex]
Then, applying the quadratic formula:
[tex]\[ x = \frac{7 \pm \sqrt{65}}{2}. \][/tex]
This gives us the following two roots:
[tex]\[ x_1 = \frac{7 - \sqrt{65}}{2}, \][/tex]
[tex]\[ x_2 = \frac{7 + \sqrt{65}}{2}. \][/tex]
Therefore, the solutions to the quadratic equation are:
[tex]\[ x_1 = \frac{7 - \sqrt{65}}{2}, \text{ and } x_2 = \frac{7 + \sqrt{65}}{2}. \][/tex]
Hence, the correct choice from the given multiple options is:
[tex]\[ \boxed{\frac{7 - \sqrt{65}}{2}, \frac{7 + \sqrt{65}}{2}}. \][/tex]
