Question

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

Short answer

The solution of the quadratic equation is$b=\frac{-1\pm \sqrt{19}}{5}$

Step-by-step solution

Given information

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

Identify the values of x,y,z

On comparing with standard equation we get$x=5,y=2,z=4$

Write quadratic formula and substitute the values of  x,y,z

The quadratic formula can be given as

$b=\frac{-y\pm \sqrt{y^2-4xz}}{2x}$

On substituting the values we get

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

Simplify and also find the common factor and remove it 

On simplifying we get

$$\begin{gathered} b=\frac{-2\pm \sqrt{4-80}}{10} \\ b=\frac{-2\pm \sqrt{-76}}{10} \\ b=\frac{-2\pm \sqrt{76}i}{10} \end{gathered}$$

Now we find the common factor and remove it

$$\begin{gathered} b=\frac{-2\pm 2\sqrt{19}i}{10} \\ b=\frac{2(-1\pm \sqrt{19}i)}{10} \\ b=\frac{-1\pm \sqrt{19}i}{5} \end{gathered}$$

Rewrite in standard form

We rewrite in standard form

$b=\frac{-1}{5}\pm \frac{\sqrt{19}}{5}i$

Conclusion

The solution of the above quadratic equation is$b=\frac{-1}{5}\pm \frac{\sqrt{19}}{5}i$