Question

1. $x^2 + 12x = -8$

Explanation:

Step1: Rearrange the equation

First, we rewrite the given equation \(x^{2}+12x = - 8\) in standard quadratic form \(ax^{2}+bx + c=0\) by adding 8 to both sides. So we get \(x^{2}+12x + 8=0\).

Step2: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\), the solutions are given by the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Here, \(a = 1\), \(b=12\) and \(c = 8\).

First, we calculate the discriminant \(\Delta=b^{2}-4ac=(12)^{2}-4\times1\times8=144 - 32=112\).

Then, we find the square root of the discriminant: \(\sqrt{112}=\sqrt{16\times7} = 4\sqrt{7}\).

Now, we substitute the values of \(a\), \(b\) and \(\Delta\) into the quadratic formula:

\(x=\frac{-12\pm4\sqrt{7}}{2\times1}=\frac{-12\pm4\sqrt{7}}{2}=- 6\pm2\sqrt{7}\)

Answer:

The solutions of the equation \(x^{2}+12x=-8\) are \(x=-6 + 2\sqrt{7}\) and \(x=-6-2\sqrt{7}\)