Exercise 950Algebra › Equations and radicalsL1Added 3 Nov 2026

A radical equation with two square roots

#radical#two roots#conditions

Statement

Solve in $\mathbb{R}$ the equation
$\sqrt{x^2-4x+4}=x-1.$

Hint

The LHS is $|x-2|$ . Discuss cases.

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Simplify. $x^2-4x+4=(x-2)^2$ , so LHS $=|x-2|$ .

Condition. RHS $x-1\ge0$ , i.e. $x\ge1$ .

Case $x\ge2$ . $|x-2|=x-2$ , so $x-2=x-1$ , i.e. $-2=-1$ : impossible.

Case $1\le x<2$ . $|x-2|=2-x$ , so $2-x=x-1$ , i.e. $2x=3$ , $x=3/2$ .

Conclusion. $x=3/2$ .