Question

1. $x^2 + 5 = -5x$

Explanation:

Step1: Rearrange the equation

We start with the equation \(x^{2}+5 = - 5x\). To solve a quadratic equation, we first write it in standard form \(ax^{2}+bx + c=0\). So, we add \(5x\) to both sides of the equation:
\(x^{2}+5x + 5=0\)

Step2: Identify coefficients

For a quadratic equation \(ax^{2}+bx + c = 0\), here \(a = 1\), \(b = 5\), and \(c = 5\).

Step3: Use the quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substitute the values of \(a\), \(b\), and \(c\) into the formula:
First, calculate the discriminant \(\Delta=b^{2}-4ac=(5)^{2}-4\times1\times5=25 - 20 = 5\)
Then, \(x=\frac{-5\pm\sqrt{5}}{2\times1}=\frac{-5\pm\sqrt{5}}{2}\)

Answer:

The solutions are \(x=\frac{-5 + \sqrt{5}}{2}\) and \(x=\frac{-5-\sqrt{5}}{2}\)