Question 85463
{{{sqrt(x+6) + sqrt(2-x) = 4}}} Start with the given equation


{{{sqrt(x+6)=4 - sqrt(2-x)}}} Subtract {{{sqrt(2-x)}}} from both sides


{{{x+6=(4 - sqrt(2-x))^2}}}  Square both sides


{{{x+6=16 - 8sqrt(2-x)+2-x}}} FOIL the right side


{{{x+6-16-2+x= - 8sqrt(2-x)}}}Get everything but the term {{{- 8sqrt(2-x)}}} to the left side


{{{2x-12= - 8sqrt(2-x)}}} Combine like terms on the left side


{{{(2x-12)^2= 64*(2-x)}}} Square both sides


{{{4x^2-48x+144= 64*(2-x)}}} FOIL the left side


{{{4x^2-48x+144= 128-64x}}} Distribute


{{{4x^2+16x+16= 0}}} Get everything to one side


{{{4(x^2+4x+4)= 0}}} Factor out a 4


{{{4(x+2)(x+2)= 0}}} Factor the expression in the parenthesis


So we have an answer of


{{{x=-2}}}


Check:

{{{sqrt(-2+6) + sqrt(2--2) = 4}}} Plug in {{{x=-2}}}


{{{sqrt(4) + sqrt(4) = 4}}}


{{{2+2 = 4}}}


{{{4=4}}} solution works