Answer:
The solutions to the quadratic equation involving square roots are:
[tex]x=2\sqrt{7}i+8,\:x=-2\sqrt{7}i+8[/tex]
Step-by-step explanation:
Given the equation
[tex]\frac{1}{4}\left(x-8\right)^2+8=1[/tex]
subtract 8 from both sides
[tex]\frac{1}{4}\left(x-8\right)^2+8-8=1-8[/tex]
[tex]\frac{1}{4}\left(x-8\right)^2=-7[/tex]
[tex]\left(x-8\right)^2=-28[/tex]
[tex]\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)}[/tex]
solving
[tex]x-8=\sqrt{-28}[/tex]
[tex]x-8=\sqrt{-1}\sqrt{28}[/tex]
[tex]x-8=\sqrt{28}i[/tex] ∵ [tex]\sqrt{-1}=i[/tex]
[tex]x=2\sqrt{7}i+8[/tex]
similarly solving
[tex]x-8=-\sqrt{-28}[/tex]
[tex]x-8=-2\sqrt{7}i[/tex]
[tex]x=-2\sqrt{7}i+8[/tex]
Therefore, the solutions to the quadratic equation involving square roots are:
[tex]x=2\sqrt{7}i+8,\:x=-2\sqrt{7}i+8[/tex]
