Question

Choose a method to solve the quadratic equation. Explain your choice. $$x^{2}-5 x-1=0$$

Short answer

The solutions to the quadratic equation \(x^{2}-5x-1=0\) are \(x1 = \frac{5 + \sqrt{29}}{2}\) and \(x2 = \frac{5 - \sqrt{29}}{2}\).

Step-by-step solution

Identify the coefficients

The quadratic equation is of the form \( ax^{2} + bx + c = 0 \). For the given equation \(x^{2}-5x-1=0\), the coefficients are \(a = 1\), \(b = -5\) and \(c = -1\).

Insert the coefficients into the quadratic formula

Enter the values into the quadratic formula. This is \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Therefore, we get \(\frac{5 \pm \sqrt{(-5)^2-4*1*(-1)}}{2*1}\).

Simplify the equation

Simplify the square root term under the equation \(\sqrt{(-5)^2-4*1*(-1)} = \sqrt{25 + 4} = \sqrt{29}\). This means the equation becomes \(\frac{5 \pm \sqrt{29}}{2}\).

Calculate the two solutions

The solutions are then \(x1 = \frac{5 + \sqrt{29}}{2}\) and \(x2 = \frac{5 - \sqrt{29}}{2}\).