Question

Solve the equation using square roots. Write your answer in simplest form. Question in picture

Answer 1

Answer:

The solutions to the quadratic equation involving square roots are:

$x=2\sqrt{7}i+8,\:x=-2\sqrt{7}i+8$

Step-by-step explanation:

Given the equation

$\frac{1}{4}\left(x-8\right)^2+8=1$

subtract 8 from both sides

$\frac{1}{4}\left(x-8\right)^2+8-8=1-8$

$\frac{1}{4}\left(x-8\right)^2=-7$

$\left(x-8\right)^2=-28$

$\mathrm{For\:}\left(g\left(x\right)\right)^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}g\left(x\right)=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}$

solving

$x-8=\sqrt{-28}$

$x-8=\sqrt{-1}\sqrt{28}$

$x-8=\sqrt{28}i$       ∵ $\sqrt{-1}=i$

$x=2\sqrt{7}i+8$

similarly solving

$x-8=-\sqrt{-28}$

$x-8=-2\sqrt{7}i$

$x=-2\sqrt{7}i+8$

Therefore, the solutions to the quadratic equation involving square roots are:

$x=2\sqrt{7}i+8,\:x=-2\sqrt{7}i+8$