Surd Equations

A surd equation has the variable under a square root. You remove the root by squaring, but squaring can create false answers, so checking every answer is part of the method.

Isolate the surd and square both sides

Get the surd on its own on one side.
Square both sides completely.
Solve the equation that results.
Check every answer in the original equation.

Example. $x - 2 = 4$

$x - 2 = 16 \Rightarrow x = 18$

Check: $16 = 4$ ✓

Example. $2 x - x + 1 = 1$

$2 x - 1 = x + 1 \Rightarrow 4 x^{2} - 4 x + 1 = x + 1 \Rightarrow 4 x^{2} - 5 x = 0$

So $x = 0$ or $x = \frac{5}{4}$ , and now both must be checked.

Check for extraneous roots

An extraneous root satisfies the squared equation but not the original one.

Check the example above.

$x = 0$ : $0 - 1 = - 1 \neq = 1$ ✗
$x = \frac{5}{4}$ : $\frac{5}{2} - \frac{9}{4} = \frac{5}{2} - \frac{3}{2} = 1$ ✓

So $x = \frac{5}{4}$ only.

Quick test before solving. $\dots \geq 0$ , so after isolating, the other side must also be $\geq 0$ . In $3 - x = 1 - x$ , you need $1 - x \geq 0$ , so $x \leq 1$ . Any answer above 1 is rejected.

Key terms

Surd equation: An equation with the variable under a root sign.
Extraneous root: A false solution introduced by squaring.
Isolate: Get one term alone on one side of the equation.

Common mistakes

Squaring before the surd is isolated.
Squaring each term separately instead of the whole side.
Not checking answers in the original equation.
Rejecting a negative answer without checking it.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

Solve for $x$ : $x - 2 = 4$

Worked solution

Hidden - try the question first

Solution

$x - 2 = 16$
$x = 18$
Check: $16 = 4$ ✓

Question 2

5 marks total

routine

Solve for $x$ : $x + 6 = x$

Worked solution

Hidden - try the question first

Solution

$x - 2 = 16$
$x^{2} - x - 6 = 0 \Rightarrow (x - 3) (x + 2) = 0$
$x = 3$ or $x = - 2$
Check $x = - 2$ : $4 = 2 \neq = - 2$ , so reject
$x = 3$

Question 3

6 marks total

complex

Solve for $x$ : $2 x - x + 1 = 1$

Worked solution

Hidden - try the question first

Solution

$x - 2 = 16$
$4 x^{2} - 4 x + 1 = x + 1$
$x (4 x - 5) = 0$
$x = 0$ or $x = \frac{5}{4}$
Check $x = 0$ : $- 1 \neq = 1$ , so reject
$x = \frac{5}{4}$

Question 4

8 marks total

problem-solving

(a) Solve for $x$ : $3 - x + x = 1$ (5)

(b) Before solving, how could you have known that any answer greater than 1 must be rejected? (3)

Worked solution

Hidden - try the question first

Solution

(a) $3 - x = 1 - x$
$3 - x = 1 - 2 x + x^{2}$
$x^{2} - x - 2 = 0 \Rightarrow (x - 2) (x + 1) = 0$
Check $x = 2$ : $1 + 2 = 3 \neq = 1$ , so reject
$x = - 1$ ( $4 - 1 = 1$ ✓)
(b) A square root is never negative
so $1 - x \geq 0$
which means $x \leq 1$

FAQs

Why do I have to check surd equation answers?

Squaring can introduce answers that do not satisfy the original equation. Only checking shows which ones are real solutions.

How many surds will the equation have in Grade 11?

One surd term. Isolate it before squaring.

Is this topic examined in Grade 12 as well?

Yes. Surd equations leading to quadratics are part of the Grade 12 algebra section.

Isolate the surd and square both sides

Check for extraneous roots

Key terms

Common mistakes

Practice questions

FAQs

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