Question

$x^2 - 5x = 8$
select the correct response
$x = \frac{5 + \sqrt{57}}{2}$
$x = \frac{5 - \sqrt{57}}{2}$
$x = \frac{-5 + \sqrt{33}}{2}$
$x = \frac{-5 - \sqrt{33}}{2}$
$x = \frac{-5 + \sqrt{57}}{2}$
$x = \frac{-5 - \sqrt{57}}{2}$
$x = \frac{5 + \sqrt{33}}{2}$
$x = \frac{5 - \sqrt{33}}{2}$

Explanation:

Step1: Rewrite the equation in standard form

The given equation is \(x^{2}-5x = 8\). Subtract 8 from both sides to get it in the standard quadratic form \(ax^{2}+bx + c=0\). So we have \(x^{2}-5x - 8=0\), where \(a = 1\), \(b=- 5\), and \(c=-8\).

Step2: Apply the quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substitute \(a = 1\), \(b=-5\), and \(c = - 8\) into the formula. First, calculate the discriminant \(b^{2}-4ac=(-5)^{2}-4\times1\times(-8)=25 + 32=57\). Then, \(x=\frac{-(-5)\pm\sqrt{57}}{2\times1}=\frac{5\pm\sqrt{57}}{2}\), which gives the two solutions \(x=\frac{5 + \sqrt{57}}{2}\) and \(x=\frac{5-\sqrt{57}}{2}\).

Answer:

\(x=\frac{5 + \sqrt{57}}{2}\), \(x=\frac{5-\sqrt{57}}{2}\) (corresponding to the first two options: \(x=\frac{5 + \sqrt{57}}{2}\) and \(x=\frac{5-\sqrt{57}}{2}\))