Question

which are the solutions of the quadratic equation?
x² = 7x + 4
\frac{-7 - \sqrt{65}}{2}, \frac{-7 + \sqrt{65}}{2}
7, 0
-7, 0

Explanation:

Step1: Rewrite to standard quadratic form

$x^2 - 7x - 4 = 0$

Step2: Identify coefficients $a,b,c$

For $ax^2+bx+c=0$, $a=1$, $b=-7$, $c=-4$

Step3: Apply quadratic formula

Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute values:
$x=\frac{-(-7)\pm\sqrt{(-7)^2-4(1)(-4)}}{2(1)}$

Step4: Simplify the expression

Calculate discriminant: $(-7)^2-4(1)(-4)=49+16=65$
So $x=\frac{7\pm\sqrt{65}}{2}$
(Note: The first option has a sign error; the correct solutions are $\frac{7-\sqrt{65}}{2}, \frac{7+\sqrt{65}}{2}$, but if we assume a typo in the question's first option, it is the intended correct choice as others are invalid)

Answer:

$\frac{7-\sqrt{65}}{2}, \frac{7+\sqrt{65}}{2}$
(closest match from given options: $\frac{-7-\sqrt{65}}{2}, \frac{-7+\sqrt{65}}{2}$ contains a sign error, but the other options are incorrect; the actual correct solutions are $\frac{7\pm\sqrt{65}}{2}$)