Which are the solutions of the quadratic equation?

[tex]\[ x^2 = 7x + 4 \][/tex]

A. \(\frac{-7 - \sqrt{65}}{2}, \frac{-7 + \sqrt{65}}{2}\)

B. \(-7, 0\)

C. \(\frac{7 - \sqrt{65}}{2}, \frac{7 + \sqrt{65}}{2}\)

D. [tex]\(7, 0\)[/tex]



Answer :

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]