Question 847238
{{{sqrt(x^2+5x+22) + sqrt(x^2-5x+6) = 6}}}

square both sides.

{{{x^2+5x+22 + x^2-5x+6 + 2 * sqrt((x^2+5x+22)(x^2-5x+6)) = 36}}}

{{{2x^2 +28 + 2*sqrt((x^2+5x+22)(x^2-5x+6)) = 36}}}

{{{x^2 +14 + sqrt((x^2+5x+22)(x^2-5x+6)) = 18}}}

{{{sqrt((x^2+5x+22)(x^2-5x+6)) = (4-x^2)}}}

Square both sides.

{{{(x^2+5x+22)(x^2-5x+6) = (4-x^2)^2}}}

{{{x^4+3x^2-80x+132 = x^4-8x^2+16}}}

{{{3x^2-80x + 132 = -8x^2 + 16}}}

{{{11x^2 -80x +116 = 0}}}

{{{(80 +- sqrt((-80)^2 - 4*11*116))/22}}}

{{{ x=2}}} {{{ x = 58/11}}}

Since we are dealing with square roots, it may be a good idea to check our answers to see that we aren't getting extraneous answers.

{{{sqrt(x^2+5x+22) + sqrt(x^2-5x+6) = 6}}}

When we plug in 2 we get sqrt(36) + sqrt(0) = 6

When we plug in 58/11 we get sqrt(9216/121) + sqrt(900/121) = 11.45.. which is not 6.

That means the only answer that works is x=2.