Question

1. $2x^2 - 8 = 24x$

Explanation:

Step1: Rearrange the equation

First, we need to rearrange the given equation \(2x^{2}-8 = 24x\) into standard quadratic form \(ax^{2}+bx + c = 0\). Subtract \(24x\) from both sides:
\(2x^{2}-24x - 8=0\)
We can simplify this equation by dividing all terms by 2:
\(x^{2}-12x - 4 = 0\)

Step2: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\), the quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Here, \(a = 1\), \(b=- 12\), and \(c=-4\).
First, calculate the discriminant \(\Delta=b^{2}-4ac\). Substitute the values:
\(\Delta=(-12)^{2}-4\times1\times(-4)=144 + 16=160\)
Then, find the square root of the discriminant: \(\sqrt{\Delta}=\sqrt{160}=\sqrt{16\times10}=4\sqrt{10}\)
Now, substitute into the quadratic formula:
\(x=\frac{-(-12)\pm4\sqrt{10}}{2\times1}=\frac{12\pm4\sqrt{10}}{2}=6\pm2\sqrt{10}\)

Answer:

\(x = 6 + 2\sqrt{10}\) or \(x = 6 - 2\sqrt{10}\)