Grade 11 Maths · Algebra

Complete the square

Rewrite a quadratic in the form a(x + p)² + q by adding and subtracting a square (for example x² + 6x + 1 = (x + 3)² − 8).

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Parabola Equation In Turning Point Form | Part 2 | Grade 11 FunctionsKevinmathscience

5) Completing square gr 11 | IntroKevinmathscience

Complete Square Grade 12 Analytical GeometryKevinmathscience

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Question 6.1 · November 2025 · Paper 1 · Provincial · 3 marks

The graphs of

f (x) = - x^{2} - 6 x

and

g (x) = a x^{2} + b x

are drawn below.

g

is the reflection of

f

along the

x

-axis. A and B

(- 5; n)

are symmetrical on

f

along

x = - p

, i.e. the axes of symmetry. $Diagram for this question$

6.1 Write down the equation of

f

in the turning point form, i.e.

y = a (x + p)^{2} + q

(3)

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

More past paper questions: complete the square

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

3. Question 1.3.1 · March 2025 · Provincial · 3 marks
1.3 Given the expression: $2 x^{2} + 8 x + 10$
1.3.1 Write the above expression in the form $a (x + p)^{2} + q$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Factorise a $y = 2 (x^{2} + 4 x + 5)$
  1 mark · Complete the square
2. ✓
  Complete the square $y = 2 [(x + 2)^{2} - 4 + 5]$
  1 mark · Complete the square
3. ✓
  Final form $y = 2 (x + 2)^{2} + 2$
  1 mark · Complete the square
4. Question 4.4 · November 2024 · Paper 1 · Provincial · 3 marks
Given the number pattern $24$ ; $10$ ; $0$ ; $- 6$ ; ...
4.4 Determine the smallest value of the quadratic number pattern.(3)
Show the memo (3 ticks)Memo
1. ✓
  Value of $n$ $T_{n} = 2 n^{2} - 20 n + 42$
  $n = \frac{- b}{2 a} = \frac{- ( - 20 )}{2 ( 2 )} = 5$
  1 mark · Use a quadratic general term
2. ✓
  Substitution $T_{5} = 2 (5)^{2} - 20 (5) + 42$
  1 mark · Use a quadratic general term
3. ✓
  Answer $= - 8$
  Therefore, the smallest value of the sequence is $- 8$
  1 mark · Use a quadratic general term
5. Question WE2 · Siyavula Grade 11 Mathematics, 5.1 Quadratic functions, Worked example 2 · 3 marks
WE2 Determine the turning point of $g (x) = 3 x^{2} - 6 x - 1$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Take out the coefficient $g (x) = 3 (x^{2} - 2 x) - 1$
  1 mark · Complete the square
2. ✓
  Completed square $g (x) = 3 ((x - 1)^{2} - 1) - 1 = 3 (x - 1)^{2} - 3 - 1 = 3 (x - 1)^{2} - 4$
  1 mark · Complete the square
3. ✓
  Turning pointFrom $g (x) = 3 (x - 1)^{2} - 4$ the turning point is $(1; - 4)$ .
  1 mark · Find turning point of a parabola
6. Question WE7 · Siyavula Grade 11 Mathematics, 2.2 Completing the square, Worked example 7 · 5 marks
WE7 Solve by completing the square: $2 x^{2} - 6 x - 10 = 0$ (5)
Show the memo (5 ticks)Memo
1. ✓
  Coefficient of x squared equal to 1Divide the entire equation by $2$ :
  $x^{2} - 3 x - 5 = 0$
  1 mark · Complete the square
2. ✓
  Add and subtract the square $(\frac{- 3}{2})^{2} = \frac{9}{4}$
  $(x^{2} - 3 x + \frac{9}{4}) - \frac{9}{4} - 5 = 0$
  1 mark · Complete the square
3. ✓
  Perfect square $(x - \frac{3}{2})^{2} - \frac{9}{4} - \frac{20}{4} = 0$
  $(x - \frac{3}{2})^{2} - \frac{29}{4} = 0$
  1 mark · Complete the square
4. ✓
  Difference of two squares $(x - \frac{3}{2})^{2} - (\frac{29}{4})^{2} = 0$
  $(x - \frac{3}{2} - \frac{29}{2}) (x - \frac{3}{2} + \frac{29}{2}) = 0$
  1 mark · Factorise a difference of squares
5. ✓
  Answer $x = \frac{3}{2} + \frac{29}{2}$ or $x = \frac{3}{2} - \frac{29}{2}$ , that is $x = \frac{3 \pm 29}{2}$
  1 mark · Solve a quadratic by factorising

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