Algebra

4 + 3*sqrt(x-4) = x

Step-by-step solution with explanation

Final Answer

and (both solutions are valid)

Step-by-step solution

Isolate the square root term

Subtract 4 from both sides so the square root expression is alone on the left. This sets us up to eliminate the radical by squaring.

Square both sides to remove radical

Squaring both sides eliminates the square root. Remember that squaring can introduce extraneous solutions, so we must check answers at the end.

Simplify both sides

The left side becomes and the right side is . Now we have a polynomial equation to solve.

Move all terms to one side

Subtract from both sides. This groups the equation nicely for factoring.

Factor out the common term

Factor out from both terms. This gives two factors whose product is zero.

Simplify and set each factor to zero

Simplify . Setting each factor equal to zero gives the two candidate solutions and .

Check both solutions in original equation

Plug each value back into . Both and satisfy the original equation, so neither is extraneous.

Understanding this problem

Learning Insight

When you square both sides of an equation, you can create extra (extraneous) solutions because squaring makes negative values look positive. That's why checking every solution in the original equation is not optional — it's a required part of solving radical equations.

Quick Tip

After isolating the radical, let as a substitution. The equation becomes , which squares to , giving very quickly.

Common Mistake

Students often forget to check for extraneous solutions and accept all algebraic answers as valid. In this problem both solutions happen to be real, but skipping the check is the habit that causes wrong answers on other radical equations.