Radical Equations

Radical equations are mathematical expressions in which the variable is found within a radical, most commonly a square root or cube root. These equations frequently arise in science, engineering, and real-world measurements involving nonlinear relationships. To solve a radical equation, the standard procedure is to isolate the radical expression and then eliminate the radical by raising each side to a power equal to the index of the radical. This process may lead to extraneous solutions—values that solve the transformed equation but not the original one—so all potential solutions must be verified by substitution.

Find the real solution(s) to the equation: 3x + 1 + 2 = x

1. Isolate the radical:

2. Square both sides (eliminating the square root):

3. Bring all terms to one side to form a standard quadratic equation:

5. Solve the quadratic using the quadratic formula:

6. Check for extraneous solutions: Only x ≈ 6.54 satisfies the original equation. Substituting x ≈ 0.46 back into the expression 3x + 1​ results in a positive value, whereas the expression x - 2 becomes negative. Since the square root is defined to yield a nonnegative result and cannot equal a negative number,
x ≈ 0.46​ is extraneous.