Chapter 10.1, Problem 98E

To determine

To find: The roots of the quadratic equation $x^2+3x-1=0$ by completing the square.

Answer to Problem 98E

The roots of the quadratic equation $x^2+3x-1=0$ are $x=\frac{-3+\sqrt{13}}{2},-\frac{3+\sqrt{13}}{2}$ .

Explanation of Solution

Given information:

The quadratic equation is $x^2+3x-1=0$ .

Calculation:

Simplify the quadratic equation by completing the square.

  $\begin{array}{c}{x}^2+3x-1=0\\ x^2+3x+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2-1=0\\ {\left(x+\frac{3}{2}\right)}^2-\frac{9}{4}-1=0\\ {\left(x+\frac{3}{2}\right)}^2-\frac{13}{4}=0\end{array}$

Now the quadratic equation gives.

  $\begin{array}{c}{\left(x+\frac{3}{2}\right)}^2-\frac{13}{4}=0\\ {\left(x+\frac{3}{2}\right)}^2=\frac{13}{4}\\ x+\frac{3}{2}=\pm \frac{\sqrt{13}}{2}\\ x=-\frac{3}{2}\pm \frac{\sqrt{13}}{2}\\ x=\frac{-3+\sqrt{13}}{2},-\frac{3+\sqrt{13}}{2}\end{array}$

So, the solution of the quadratic equation $x^2+3x-1=0$ are $x=\frac{-3+\sqrt{13}}{2},-\frac{3+\sqrt{13}}{2}$ .

Therefore, the roots of the quadratic equation $x^2+3x-1=0$ are $x=\frac{-3+\sqrt{13}}{2},-\frac{3+\sqrt{13}}{2}$ .