Surd Equations

Q: Why do surd equations give extra answers?

Squaring both sides makes $a = b$ and $a = -b$ look the same. One of the answers may come from the negative case and not satisfy the original equation.

Q: Do I lose marks if I do not check my answers?

Yes. The memo gives a mark for rejecting the extraneous root, and including it costs the final answer mark.

Q: Can the answer to a surd equation be negative?

Yes, as long as it satisfies the original equation. Only the check decides.

A surd equation has the variable under a square root. Squaring both sides removes the root but can create false answers, so checking is part of the method, not an optional extra. These questions are usually worth 4 to 6 marks.

Solving equations with one surd

Isolate the surd on one side.
Square both sides. Square the whole side, not each term.
Solve the resulting quadratic.
Check every answer in the original equation.

Example. $x - 2 x + 6 = 1$

$x - 1 = 2 x + 6$

$x^{2} - 2 x + 1 = 2 x + 6$

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

Possible answers: $x = 5$ or $x = - 1$ . Now check both.

Squaring $(x - 1)$ gives $x^{2} - 2 x + 1$ , not $x^{2} + 1$ . The middle term is where many marks are lost.

Checking for extraneous roots

An extraneous root is a value that solves the squared equation but not the original one. It appears because $a^{2} = b^{2}$ is true when $a = - b$ as well as when $a = b$ .

Check the example above.

$x = 5$ : $5 - 16 = 5 - 4 = 1$ ✓
$x = - 1$ : $- 1 - 4 = - 3 \neq = 1$ ✗

So $x = 5$ only.

A quick filter: $\dots \geq 0$ always, so the other side of the isolated equation must also be $\geq 0$ . In $x - 1 = 2 x + 6$ , you need $x \geq 1$ , which rules out $x = - 1$ immediately.

Key terms

Surd: A root that cannot be simplified to a rational number, such as $2$ .
Extraneous root: A value produced by squaring that does not satisfy the original equation.
Isolate: Get the surd alone on one side of the equation.

Common mistakes

Squaring before isolating the surd.
Squaring term by term: $(x - 1)^{2} \neq = x^{2} - 1$ .
Not checking answers, or checking in the squared equation instead of the original.
Rejecting a valid answer because it is negative; only the check decides.

Practice questions

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

Question 1

4 marks total

routine

Solve for $x$ : $x + 7 = x + 1$

Worked solution

Hidden - try the question first

Solution

$x + 7 = x^{2} + 2 x + 1$
$x^{2} + x - 6 = 0 \Rightarrow (x + 3) (x - 2) = 0$
Check $x = - 3$ : $4 = 2 \neq = - 2$ , so reject
$x = 2$

Question 2

5 marks total

routine

Solve for $x$ : $x - 2 x + 6 = 1$

Worked solution

Hidden - try the question first

Solution

$x + 7 = x^{2} + 2 x + 1$
$x^{2} - 2 x + 1 = 2 x + 6$
$(x - 5) (x + 1) = 0$
Check $x = - 1$ : $- 1 - 2 = - 3 \neq = 1$ , so reject
$x = 5$

Question 3

6 marks total

complex

Solve for $x$ : $3 x + 1 + 3 = x$

Worked solution

Hidden - try the question first

Solution

$x + 7 = x^{2} + 2 x + 1$
$x^{2} - 2 x + 1 = 2 x + 6$
$(x - 5) (x + 1) = 0$
$(x - 1) (x - 8) = 0$
Check $x = 1$ : $4 + 3 = 5 \neq = 1$ , so reject
Check $x = 8$ : $25 + 3 = 8$ ✓, so $x = 8$

Question 4

8 marks total

problem-solving

(a) Solve for $x$ : $2 x - 1 = x - 2$ (5)

(b) The graphs of $y = 2 x - 1$ and $y = x - 2$ are drawn on the same axes. Explain why they meet only once. (1)

Worked solution

Hidden - try the question first

Solution

(a) $2 x - 1 = x^{2} - 4 x + 4$
$x^{2} - 6 x + 5 = 0$
$(x - 1) (x - 5) = 0$
Check $x = 1$ : $1 = 1 \neq = - 1$ , so reject
$x = 5$
(b) At $x = 1$ the line is below the $x$ -axis but the root graph is above it, so $x = 1$ is not a meeting point; they meet only at $x = 5$
(c) The surd is defined for $x \geq \frac{1}{2}$ , and the root graph is above the line to the left of $x = 5$
$\frac{1}{2} \leq x < 5$

FAQs

Why do surd equations give extra answers?

Squaring both sides makes $a = b$ and $a = - b$ look the same. One of the answers may come from the negative case and not satisfy the original equation.

Do I lose marks if I do not check my answers?

Yes. The memo gives a mark for rejecting the extraneous root, and including it costs the final answer mark.

Can the answer to a surd equation be negative?

Yes, as long as it satisfies the original equation. Only the check decides.

Solving equations with one surd

Checking for extraneous roots

Key terms

Common mistakes

Practice questions

FAQs

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