Question 1152365
{{{sqrt(10-x)+sqrt(3+x)+2sqrt(30+7x-x^2)=17}}}


{{{sqrt(10-x)+sqrt(3+x)=17-2sqrt(30+7x-x^2)}}}......square both sides


{{{(sqrt(10-x)+sqrt(3+x))^2=(17-2sqrt(30+7x-x^2))^2}}}


{{{(10-x)+2sqrt(10-x)sqrt(3+x)+3+x=289-68sqrt(30+7x-x^2)+4(30+7x-x^2)}}}


{{{2sqrt((10-x)(3+x))+13=289-68sqrt(30+7x-x^2)+120+28x-4x^2}}}


{{{2sqrt((10-x)(3+x))=409+28x-4x^2-68sqrt(30+7x-x^2)-13}}}


{{{2sqrt((10-x)(3+x))=396+28x-4x^2-68sqrt(30+7x-x^2) }}}.........square both sides

{{{(2sqrt((10-x)(3+x)))^2=(396+28x-4x^2-68sqrt(30+7x-x^2))^2}}}


{{{120+28x-4x^2=16 x^4 - 224 x^3 + 544 sqrt(-x^2 + 7 x + 30)* x^2 - 7008 x^2 - 3808 sqrt(-x^2 + 7 x + 30)* x - 53856 sqrt(-x^2 + 7 x + 30) + 54544 x + 295536}}}

{{{120+28x-4x^2-16 x^4 + 224 x^3+7008 x^2-54544 x -295536=544 sqrt(-x^2 + 7 x + 30)* x^2- 3808 sqrt(-x^2 + 7 x + 30)* x - 53856 sqrt(-x^2 + 7 x + 30) }}}


{{{-16 x^4 + 224 x^3 + 7004 x^2 - 54516 x - 295416=(544x^2- 3808x) sqrt(-x^2 + 7 x + 30) - 53856 sqrt(-x^2 + 7 x + 30)}}} .........square both sides

{{{(-16 x^4 + 224 x^3 + 7004 x^2 - 54516 x - 295416)^2=((544x^2- 3808x) sqrt(-x^2 + 7 x + 30) - 53856 sqrt(-x^2 + 7 x + 30) )^2}}}
 
{{{16 (4 x^4 - 56 x^3 - 1751 x^2 + 13629 x + 73854)^2 = -295936 (x - 10) (x + 3) (x^2 - 7 x - 99)^2}}}

{{{256 x^8 - 7168 x^7 - 173952 x^6 + 4882304 x^5 + 34086160 x^4 - 896006496 x^3 - 1166193072 x^2 + 32209797312 x + 87270613056 = -295936 x^6 + 6214656 x^5 + 23970816 x^4 - 843121664 x^3 - 1352131584 x^2 + 32608300032 x + 87014062080}}}

{{{256 x^8 - 7168 x^7 - 173952 x^6 + 4882304 x^5 + 34086160 x^4 - 896006496 x^3 - 1166193072 x^2 + 32209797312 x + 87270613056 +295936 x^6- 6214656 x^5 -23970816 x^4 + 843121664 x^3 +1352131584 x^2 - 32608300032 x - 87014062080=0}}}

{{{56 x^8 - 7168 x^7 +121984 x^6-1332352 x^5+ 10115344 x^4 -52884832 x^3+185938512 x^2 - 398502720 x+256550976=0}}}..........factor

{{{16 (x - 6) (x - 1) (4 (x - 7) x + 409) ((x - 7) x (4 (x - 7) x + 297) + 6534) = 0}}}

real solutions:

{{{(x - 6)=0}}} => {{{x=6 }}}
{{{(x - 1)=0}}} => {{{x=1}}}


{{{(4 (x - 7) x + 409) ((x - 7) x (4 (x - 7) x + 297) + 6534)=0}}}->   complex solutions:

{{{x = 7/2 + 3i*sqrt(10)}}}

{{{x = 7/2 - 3i*sqrt(10)}}}

{{{x = (1/4 )(14 - sqrt(-398 - 66i*sqrt(15)))}}}

{{{x = (1/4 )(14 +sqrt(-398 - 66i*sqrt(15)))}}}

{{{x = (1/4 )(14 - sqrt(-398 + 66i*sqrt(15)))}}}