How do you solve sqrt( x + 1) =x - 5x+1=x5?

1 Answer
Apr 22, 2016

Square both sides, solve the resulting quadratic, then check the solutions to find valid solution:

x = 8x=8

Explanation:

First square both sides, noting that this may introduce spurious solutions, to get:

x+1 = x^2-10x+25x+1=x210x+25

Subtract x+1x+1 from both sides to get:

0 = x^2-11x+24 = (x-3)(x-8)0=x211x+24=(x3)(x8)

So x = 3x=3 or x = 8x=8

The solution x=3x=3 of this quadratic is spurious since x-5 = -2 < 0x5=2<0, so does not match the positive square root in the original equation.

The solution x=8x=8 is a valid solution of the original equation:

sqrt(8+1) = sqrt(9) = 3 = 8 - 58+1=9=3=85