How do i find the extraneous value?

Question

√(x+6) -x =4

Answer 1

The equation is √(x+6) -x =4 .

√(x+6) -x = 4

Add x to each side .

√(x+6) -x + x = 4 + x

√(x+6) = 4 + x

Now square on both sides .

[√(x+6)]² = (4 + x)²

x + 6 = 16 + x² + 8x

Now subtract x + 6 from each side .

x + 6 - (x + 6) = 16 + x² + 8x  - (x + 6)

0 = 16 + x² + 8x  - x - 6

0 = 10 + x² + 7x  

10 + x² + 7x  = 0

 x² + 7x +10  = 0

 x² + 5x +2x +10  = 0

x(x+5) +2(x+5) = 0

(x+5)(x+2) = 0

x+5 = 0 and x+2 = 0

x = -5 and x = -2 

So the solutions are x = -5 and x = -2 .

check for x = -5

√(-5+6) -(-5) = 4

√(1) +5 = 4

1+5 ≠ 4

Now check for x = -2

√(-2+6) -(-2) = 4

√(4) +2 = 4

2+2 = 4

So the real root for √(x+6) -x =4  is  x = -2 .

So the extraneous value is x = -5 .