Question

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

Answer 1

The solutions to the quadratic equation involving square roots are:

`x=2√(7)i+8,\:x=-2√(7)i+8`

Explanation:

Given the equation

`(1)/(4)\left(x-8\right)^2+8=1`

subtract 8 from both sides

`(1)/(4)\left(x-8\right)^2+8-8=1-8`

`(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)=√(f\left(a\right)),\:\:-√(f\left(a\right))`

solving

`x-8=√(-28)`

`x-8=√(-1)√(28)`

`x-8=√(28)i` ∵
`√(-1)=i`

`x=2√(7)i+8`

similarly solving

`x-8=-√(-28)`

`x-8=-2√(7)i`

`x=-2√(7)i+8`

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

`x=2√(7)i+8,\:x=-2√(7)i+8`