Question

Solve by using the Quadratic Formula:$5n^2+4n-4=0$

Short answer

The solution of the quadratic equation is$n=\frac{-2\pm 2\sqrt{6}}{5}$

Step-by-step solution

Given information

We are given a quadratic equation$5n^2+4n-4=0$

Identify the values of a,b,c

On comparing with standard equation we get$a=5,b=4,c=-4$

Write quadratic formula and substitute the values of a,b,c

The quadratic formula can be given as

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

On substituting the values we get

$n=\frac{-4\pm \sqrt{16-4\left(5\right)(-4)}}{2\left(5\right)}$

Simplify

On simplifying we get

$$\begin{gathered} n=\frac{-4\pm \sqrt{16+80}}{10} \\ n=\frac{-4\pm 4\sqrt{6}}{10} \end{gathered}$$

 Factor out and remove the common factor

We get

$$\begin{gathered} n=\frac{2(-2\pm 2\sqrt{6})}{10} \\ n=\frac{-2\pm 2\sqrt{6}}{5} \end{gathered}$$

Conclusion

The root of the equation is $n=\frac{-2\pm 2\sqrt{6}}{5}$