Question

which are the solutions of the quadratic equation?
x² = 9x + 6
options:
\\(\frac{-9 - \sqrt{105}}{2}, \frac{-9 + \sqrt{105}}{2}\\)
\\(\frac{-9 - \sqrt{57}}{2}, \frac{-9 + \sqrt{57}}{2}\\)
\\(\frac{9 - \sqrt{105}}{2}, \frac{9 + \sqrt{105}}{2}\\)
\\(\frac{9 - \sqrt{57}}{2}, \frac{9 + \sqrt{57}}{2}\\)

Explanation:

Step1: Rewrite the equation in standard form

First, we rewrite the quadratic equation \(x^{2}=9x + 6\) in the standard form \(ax^{2}+bx + c = 0\). Subtract \(9x\) and \(6\) from both sides:
\(x^{2}-9x - 6=0\)
Here, \(a = 1\), \(b=- 9\), and \(c=-6\).

Step2: Apply the quadratic formula

The quadratic formula for the solutions of a quadratic equation \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
Substitute \(a = 1\), \(b=-9\), and \(c = - 6\) into the formula:
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-9)^{2}-4\times1\times(-6)=81 + 24=105\)
Then, \(x=\frac{-(-9)\pm\sqrt{105}}{2\times1}=\frac{9\pm\sqrt{105}}{2}\)
So the solutions are \(\frac{9-\sqrt{105}}{2}\) and \(\frac{9+\sqrt{105}}{2}\)

Answer:

\(\frac{9 - \sqrt{105}}{2},\frac{9+\sqrt{105}}{2}\) (corresponding to the third option: \(\boldsymbol{\frac{9 - \sqrt{105}}{2},\frac{9+\sqrt{105}}{2}}\))