√(3 -x) = x + 3 square both sides
(√(3 -x))² = (x + 3)²
3 - x = (x + 3)(x + 3)
3 - x = x(x +3) + 3(x + 3)
3 - x = x² + 3x + 3x + 9
3 - x = x² + 6x + 9
0 = x² + 6x +x + 9 -3
0 = x² + 7x + 6
x² + 7x + 6 = 0 Solve the quadratic function. +6 = +6 * +1
x² + x + 6x + 6 = 0
x(x +1) + 6(x +1) = 0
(x+1)(x + 6) = 0
x +1 = 0 or x + 6 = 0
x = 0 -1, or x = 0 -6, x = -1, or -6.
Check:
√(3 -x) = x + 3 √(3 -x) = x + 3
x = -1 x = -6
√(3 - -1) = -1 + 3 √(3 - -6) = -6 + 3
√(3 +1) = 3 -1 √(3 +6) = 3 - 6
√4 = 3 -1 √9 = -3
2 = 2 Square root of 9 is also -3,
LHS = RHS
Left Hand side (LHS) = Right Hand Side (RHS)
So the solutions x = -1, and x = -6 are correct for the surd equation.