Question 963997
{{{sqrt(x^2+4x-21) +sqrt(x^2-x-6)=sqrt(6x^2 -5x-39)}}}
{{{sqrt((x+7)(x-3)) +sqrt((x-3)(x+2))=sqrt((6x+13)(x-3))}}}
Factor out the {{{sqrt(x-3)}}} from all terms.
{{{sqrt(x+7) +sqrt(x+2)=sqrt(6x+13)}}}
Square both sides,
{{{(x+7)+2sqrt((x+7)(x+2))+(x+2)=6x+13}}}
{{{2x+9+2sqrt((x+7)(x+2))+(x+2)=6x+13}}}
{{{2sqrt((x+7)(x+2))=4x+4}}}
{{{2sqrt((x+7)(x+2))=4(x+1)}}}
{{{sqrt((x+7)(x+2))=2(x+1)}}}
Square both sides,
{{{(x+7)(x+2)=4(x^2+2x+1)}}}
{{{x^2+9x+14=4x^2+8x+4}}}
{{{3x^2+9x+14=4x^2+8x+4}}}
{{{3x^2-x-10=0}}}
{{{(x-2)(3x+5)=0}}}
Remember we factored out the {{{sqrt(x-3)}}} before so that's still part of the solution. 
We only removed it so that the calculation wouldn't be messier.
{{{sqrt(x-3)(x-2)(3x+5)=0}}}
Three possible solutions:
{{{sqrt(x-3)=0}}}
{{{x-3=0}}}
{{{x=3}}}
Verifying,
{{{sqrt(3^2+4(3)-21) +sqrt(3^2-3-6)=sqrt(6(3)^2 -5(3)-39)}}}
{{{sqrt(9+12-21) +sqrt(9-9)=sqrt(54 -15-39)}}}
{{{0+0=0}}}
True, good solution.
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{{{x-2=0}}}
{{{x=2}}}
Verifying,
{{{sqrt(2^2+4(2) -21) +sqrt((2)^2-(2)-6)=sqrt(6(2)^2 -5(2)-39)}}}
{{{sqrt(4+8 -21) +sqrt(4-2-6)=sqrt( 24-5(2)-39)}}}
{{{sqrt(-9) +sqrt(-4)=sqrt( -25)}}}
Leads to a negative square root so not a real solution.
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{{{3x+5=0}}}
{{{3x=-5}}}
{{{x= (-5/3) }}}
Verifying,
{{{sqrt((-5/3)^2+4(-5/3)-21) +sqrt((-5/3)^2-(-5/3)-6)=sqrt(6(-5/3)^2 -5(-5/3)-39)}}}
{{{sqrt(25/9-20/3-21) +sqrt(25/9+5/3-6)=sqrt(50/3+25/3 -39)}}}
{{{sqrt(-224/9) +sqrt(-14/9)=sqrt(-14)}}}
Leads to a negative square root so not a real solution.
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Only one real solution, {{{x=3}}}