Technique

Square both sides to clear a radical

Algebra • Equations

Statement

A = B

then

A^{2} = B^{2}

, so squaring both sides of an equation removes a square root from it. The converse fails:

A^{2} = B^{2}

allows

A = - B

as well, so the squared equation can have roots the original does not, and every root obtained must be substituted back into the original equation before it is accepted.

How to use it

1Isolate the radical on one side first, so that squaring clears it in one step instead of producing a new cross term.
2Square the whole of each side, not term by term: $(1 + x)^{2}$ is $1 + 2 x + x$ , not $1 + x$ .
3Test every root in the original equation and discard the ones that fail; this is a required step, not a check.
4A root is usually lost to a sign: it satisfies $A = - B$ rather than $A = B$ .

Worked example

Solve

x + 7 = x - 5

Step-by-step

1
$x + 7 = (x - 5)^{2} = x^{2} - 10 x + 25$
Why: The radical stands alone, so squaring both sides clears it at once.
2
$x^{2} - 11 x + 18 = 0 \Rightarrow (x - 9) (x - 2) = 0$
3
$x = 9$ : $16 = 4 = 9 - 5$ , accepted. $x = 2$ : $9 = 3$ but $2 - 5 = - 3$ , rejected.
Why: The root $x = 2$ satisfies $A = - B$ , which squaring allowed in; only $x = 9$ solves the original equation.

Answer

x = 9

Watch out

Squaring never loses a root, so the original equation's solutions are always among the squared equation's; the danger is entirely in the other direction. A radical equal to a negative quantity has no solution at all, and squaring hides that - which is why the substitution back is compulsory.

Builds on

denotes the non-negative root Square of a sum and of a difference

Also known as

squaring both sides, square both sides of an equation, clear a radical, remove the square root, extraneous roots from squaring, irrational equation

Square both sides to clear a radical

Statement

Worked example

Appears in past papers

2020 (II)

2020 (I)

2019 (II)

2019 (I)