Question

Find the zeros by using the quadratic formula and tell whether the solutions are real or imaginary. F(x)=8x^2–4x+10

Answer 1

We have a quadratic equation and we have to find the zeros.

We can start by simplyfing:

$\begin{gathered} f(x)=8x^2-4x+10=0 \\ 2(4x^2-2x+5)=0 \\ 4x^2-2x+5=0 \end{gathered}$

Then, we apply the quadratic formula:

$\begin{gathered} x=\frac{-(-2)}{2\cdot4}\pm\frac{\sqrt[]{(-2)^2-4\cdot4\cdot5}}{2\cdot4}=\frac{2}{8}\pm\frac{\sqrt[]{4-80}}{8}=\frac{1}{4}\pm\frac{\sqrt[]{-76}}{8} \\ x=\frac{1}{4}\pm\frac{\sqrt[]{76(-1)}}{8}=\frac{1}{4}\pm\frac{\sqrt[]{76}}{8}\cdot i=\frac{1}{4}\pm\frac{\sqrt[]{4\cdot19}}{8}\cdot i=\frac{1}{4}\pm\frac{1}{4}\sqrt[]{19}i \end{gathered}$$\begin{gathered} x_1=\frac{1}{4}+\frac{1}{4}\sqrt[]{19}i\approx0.25+1.09i \\ x_2=\frac{1}{4}-\frac{1}{4}\sqrt[]{19}i\approx0.25-1.09i \end{gathered}$

The roots are imaginary.

Their values are approximately x1=0.25+1.09i and x2=0.25-1.09i.