Grade 11 Maths · Algebra

Solve a surd equation

Solve an equation with a square root by getting the root alone and squaring both sides (for example √(x + 5) = 3 gives x = 4).

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4e) Square root equations gr 11 | TryKevinmathscience

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Question 5.2 · March 2025 · Provincial · 6 marks

5.2 If

tan A = \frac{3}{12 - x}

and

12 - x = x

, Determine the value of A if

A \in [0^{\circ}; 36 0^{\circ}]

(6)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: solve a surd equation

Work each one out first, then open the memo and check every tick.

3. Question 1.1.5 · May/June 2025 · Paper 1 · National (DBE) · 4 marks
1.1 Solve for $x$ :
1.1.5 $x + 3 x + 2 = 2$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Isolate the surd $3 x + 2 = 2 - x$
  1 mark · Solve a linear equation
2. ✓
  Square both sides $9 (x + 2) = (2 - x)^{2}$
  $9 (x + 2) = 4 - 4 x + x^{2}$
  1 mark · Solve a surd equation
3. ✓
  Write in standard form and factorise $x^{2} - 13 x - 14 = 0$
  $(x - 14) (x + 1) = 0$
  1 mark · Write a quadratic in standard form
4. ✓
  Solve and check for valid solutions $x = 14$ or $x = - 1$
  Check: If $x = 14$ , $14 + 316 = 14 + 12 = 26 \neq = 2$ (Reject)
  If $x = - 1$ , $- 1 + 31 = - 1 + 3 = 2$ (Accept)
  $∴ x \neq = 14$ or $x = - 1$
  1 mark · Check solutions of a surd equation
4. Question 1.1.5 · November 2024 · Paper 1 · National (DBE) · 4 marks
1.1 Solve for $x$ :
1.1.5 $- 2 x + 4 - x = 2$ (4)
Show the memo (5 ticks)Memo
1. ✓
  Isolate the radical $- 2 x + 4 = 2 + x$
  0 marks · Solve a linear equation
2. ✓
  Square both sides $- 2 x + 4 = (2 + x)^{2}$
  $- 2 x + 4 = 4 + 4 x + x^{2}$
  1 mark · Solve a surd equation
3. ✓
  Write in standard form $x^{2} + 6 x = 0$
  1 mark · Write a quadratic in standard form
4. ✓
  Factorise and solve $x (x + 6) = 0$
  $x = 0$ or $x = - 6$
  1 mark · Solve a quadratic by factorising
5. ✓
  Validate and select correct answerCheck $x = 0$ : $4 - 0 = 2$ (Valid)
  Check $x = - 6$ : $16 - (- 6) = 10 \neq = 2$ (Invalid)
  Answer: $x = 0$ (and $x \neq = - 6$ )
  1 mark · Check solutions of a surd equation
5. Question 1.1.5 · May/June 2024 · Paper 1 · National (DBE) · 5 marks
1.1 Solve for x:
1.1.5 $\frac{x}{20 - x} = 1$ (5)
Show the memo (5 ticks)Memo
1. ✓
  Isolate the surd $x = 20 - x$
  1 mark · Solve a linear equation
2. ✓
  Square both sides $x^{2} = 20 - x$
  1 mark · Solve a surd equation
3. ✓
  Standard form $x^{2} + x - 20 = 0$
  1 mark · Write a quadratic in standard form
4. ✓
  Solve quadratic factors $x = 4$ or $x = - 5$
  1 mark · Solve a quadratic by factorising
5. ✓
  Select valid solution after check $x = 4$
  1 mark · Check solutions of a surd equation
6. Question 1.2.2 · November 2024 · Paper 1 · Provincial · 4 marks
1.2 Given: $\frac{14}{p} = p + 5$
1.2.2 Hence, or otherwise, solve for $x$ if $\frac{14}{x + 5} = x + 5 + 5$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Putting each root back: $2 = x + 5$ or $- 7 = x + 5$ Let $x + 5 = p$
  $2 = x + 5$ or $- 7 = x + 5$
  1 mark · Write a quadratic in standard form
2. ✓
  Rejecting the negative valueBut $- 7 \neq = x + 5$
  1 mark · Solve a surd equation
3. ✓
  Square both sides $x + 5 = (2)^{2}$
  1 mark · Solve a surd equation
4. ✓
  $x$ -value $x = 4 - 5$
  $x = - 1$
  1 mark · Solve a surd equation

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