A nice Math Olympiad Problem | Best Radical Simplification | Imaginary Roots |

[@lifewatery7472]

If x={R}, √(x+4) => x>=-4, √(-x-4) => x

[@ArwindSah]

Easy way to solve this equation
√(X+4) + √(-X - Y) = 4
Square both side
(√(X+4))^2 + (√(-X-Y))^2 + 2(√(X+4)×√(-X-4) =16
Or
X+4 - X -4 +2√((X+4)(-1)(X+4) = 16
Or
2√((-1)(X+4)(X+4)) =16
Or √((-1)(X+4)^2 = 8
Squaring both side
-(X+4)^2 = 64
Or
(X+4)^2 = - 64
Hence
X+4 = 8i or -8i
And
X= - 4 +8i. Or - 4 -8i
Thanks

[@heiu304]

sqrt(x+4) + sqrt(-x-4) = 4
sqrt(x+4) + sqrt(-(x+4)) = 4
sqrt(x+4) * (1+i) = 4
sqrt(x+4) = 4 / (1+i)
x+4 = (4 / (1+i))^2
= 16 / (1+i)^2
= 16 / 2i
= 8/i
= sqrt(64) / sqrt(-1)
= sqrt(-64)
= +-8i
x = -4+-8i

[@rainerzufall42]

sqrt(x+4) + sqrt((-1)(x+4) = (1 +/- i) sqrt(x+4) = 4 => x + 4 = +/- 16 / 2i { because (1 +/- i)^2 = +/- 2i }
=> x = - 4 ~/+ 8i. Check: sqrt(~/+ 8i) + sqrt(+/~ 8i) = (2 ~/+ 2i) + (2 +/~ 2i) = 2 + 2 = 4. You can see, how both solutions coordinate!

[@walterwen2975]

A nice Math Olympiad Problem: √(x + 4) + √(- x - 4) = 4; x = ?
[√(x + 4) + √(- x - 4)]² = 4², (x + 4) + (- x - 4) + 2√[(x + 4)(- x - 4)] = 16
√[(x + 4)(- x - 4)] = 8, - (x + 4)² = 8², (x + 4)² = - 8² = (8i)²; x = - 4 ± 8i
Answer check:
[√(x + 4) + √(- x - 4)]² = {√(- 4 ± 8i + 4) + √[- (- 4 ± 8i) - 4]}²
= [√(± 8i) + √(-/+ 8i)]² = (± 8i) + (-/+ 8i) + 2√[(± 8i)(-/+ 8i)]
= 2(- 8i²) = 2(8) = 16; √(x + 4) + √(- x - 4) = √16 = 4; Confirmed
Final answer:
x = - 4 + 8i or x = - 4 - 8i

[@EC4U2C_Studioz]

No real solutions.