Question

State the quadratic formula for the quadratic equation $a{x}^2+bx+c=0$, and use it to solve the equation$x^2+6x-1=0$

Short answer

The formula for the quadratic equation is$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ and the solution of the given equation is $x_1=\frac{-6+\sqrt{40}}{2},x_2=\frac{-6-\sqrt{40}}{2}$.

Step-by-step solution

Step 1. Quadratic formula method.

The quadratic formula is given by,

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Step 2. Interpretation from quadratic formula.

If $b^2-4ac>0$, the equation has two real solution.

If $b^2-4ac=0$, the equation has one solution.

If $b^2-4ac<0$, the equation has two complex solution.

Step 3. Solve the given equation.

The given equation is,

$x^2+6x-1=0$

So,

$\begin{array}{l}a=1\\ b=6\\ c=-1\end{array}$

Solve by substituting the values in quadratic formula.

$\begin{array}{c}x=\frac{-6\pm \sqrt{6^2-4\left(1\right)\left(1\right)}}{2\left(1\right)}\\ =\frac{-6\pm \sqrt{40}}{2}\end{array}$

So,

$\begin{array}{c}{x}_1=\frac{-6+\sqrt{40}}{2}\\ x_2=\frac{-6-\sqrt{40}}{2}\end{array}$

Thus, the formula for the quadratic equation is$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ and the solution of the given equation is $x_1=\frac{-6+\sqrt{40}}{2},x_2=\frac{-6-\sqrt{40}}{2}$.