Question

Cornell solved the radical equation
√
x
2
+
3
x
+
2
=
√
x
2
−
3
−
1
and found
x
=
−
2
. What is the other solution of this radical equation? Enter your answer as a fraction, or enter 0 if
x
=
−
2
is the only solution.

Answer 1

Solve sqrt(x^2+3x+2) = sqrt(x^2-3) - 1. Domain requires sqrt(x^2-3)-1 >= 0, so |x| >= 2; also x^2+3x+2 >= 0, so x <= -2 or x >= -1; together x <= -2 or x >= 2.

Isolate and square:
sqrt(x^2+3x+2)+1 = sqrt(x^2-3)
=> 2 sqrt(x^2+3x+2) = -3x - 6 (so x <= -2)
Square again:
4(x^2+3x+2) = ( -3x - 6 )^2
=> 5x^2 + 24x + 28 = 0
=> x = (-24 ± 4)/10 = -2 or -14/5.

Checking both in the original equation shows both work. The other solution is -14/5.