Determine This

Grade 11 Algebra Equations Inequalities — How to Identify the Question Type

Algebra questions in Grade 11 Paper 1 look different depending on which type is being tested. Before picking up your pen, read the question and identify which of these six types it is.

Type 1: Solving a quadratic equation

Trigger words: "Solve for x", "Find the roots", "For which values of x"

Trigger structure: If you see x² and the question says "solve" — this is it.

Do not confuse with: Simplifying (which doesn't ask you to find x).

Method (no numbers — just the steps)

  1. Factorise or use the quadratic formula
  2. Set each factor equal to zero
  3. Solve for x
Grade 11Paper 1Level 2 — Routine Procedures
2.1 Solve for x: 2x² − 5x − 3 = 0

Source: DBE Mathematics Paper 1, November 2023, Question 2.1

Common mistake

Stopping after factorising without solving for the actual value of x.

See the progression — same type, increasing difficulty

Easy
Solve for x: x² − 5x + 6 = 0

Practice question — not sourced from a past paper.

Medium
Solve for x: 3x² − 7x = 6

Practice question — not sourced from a past paper.

Hard
Solve for x (correct to TWO decimal places): 2x² + 3x − 7 = 0

Practice question — not sourced from a past paper.

Type 2: Completing the square

Trigger words: "Write in the form (x + a)² + b", "Express in the form..."

Trigger structure: "Write in the form" is the clearest signal.

Do not confuse with: Solving (unless the question also says "hence solve").

Method (no numbers — just the steps)

  1. Halve the coefficient of x
  2. Square that value
  3. Add and subtract it to keep the expression equivalent
  4. Write the result in the form (x + a)² + b
Grade 11Paper 1Level 3 — Complex Procedures
2.3 Write the expression 2x² + 12x + 11 in the form a(x + p)² + q.

Source: DBE Mathematics Paper 1, May/June 2022, Question 2.3

See the progression — same type, increasing difficulty

Easy
Write the expression x² + 6x + 5 in the form (x + p)² + q.

Practice question — not sourced from a past paper.

Medium
Write the expression 2x² + 8x + 3 in the form p(x + q)² + r.

Practice question — not sourced from a past paper.

Hard
Given: f(x) = −x² + 4x − 1 4.1 Write f(x) in the form −(x + p)² + q. 4.2 Hence, write down the maximum value of f.

Practice question — not sourced from a past paper.

Type 3: Simultaneous equations

Trigger words: "Solve simultaneously", two equations are given

Trigger structure: Two equations, two unknowns — almost always substitution: solve the linear equation for one variable, substitute into the quadratic.

Method (no numbers — just the steps)

  1. Label the equations (1) and (2)
  2. Make one variable the subject of the linear equation
  3. Substitute into the second equation
  4. Solve the resulting equation
  5. Substitute back to find the other variable
Grade 11Paper 1Level 3 — Complex Procedures
2.4 Solve the following equations simultaneously for x and y: y = x − 1 x² + y² = 25

Source: DBE Mathematics Paper 1, November 2022, Question 2.4

See the progression — same type, increasing difficulty

Easy
Solve the following equations simultaneously for x and y: y = x − 2 y = x² − 4x + 2

Practice question — not sourced from a past paper.

Medium
Solve the following equations simultaneously for x and y: x + y = 5 x² + y² = 17

Practice question — not sourced from a past paper.

Hard
Solve the following equations simultaneously for x and y: y = 2x − 3 2x² − xy + y² = 9

Practice question — not sourced from a past paper.

Type 4: Inequalities

Trigger words: "For which values of x is...", answer required in interval notation

Trigger structure: Look for >, <, ≥, ≤ symbols in the question.

Do not confuse with: Solving equations — the answer format (a range, not a single value) is different.

Method (no numbers — just the steps)

  1. Move all terms to one side
  2. Factorise (for quadratic inequalities)
  3. Determine the critical values
  4. Test intervals or sketch to determine the sign
  5. Write the answer in the required notation
Grade 11Paper 1Level 3 — Complex Procedures
3.2 Determine the value(s) of x for which: x² − x − 6 ≤ 0

Source: DBE Mathematics Paper 1, November 2021, Question 3.2

See the progression — same type, increasing difficulty

Easy
Solve for x: x² − 9 ≤ 0

Practice question — not sourced from a past paper.

Medium
Determine the value(s) of x for which: x² + 2x − 8 > 0

Practice question — not sourced from a past paper.

Hard
Solve for x: (x − 1)/(x + 2) ≥ 0

Practice question — not sourced from a past paper.

Type 5: Nature of roots (discriminant)

Trigger words: "Show that... has real/unequal/equal roots", "Determine the nature of the roots", "For which values of k will..."

Trigger structure: Involves the discriminant, Δ = b² − 4ac.

Method (no numbers — just the steps)

  1. Identify a, b and c
  2. Calculate the discriminant Δ = b² − 4ac
  3. Interpret the sign of Δ (positive = real and unequal, zero = real and equal, negative = non-real)
Grade 11Paper 1Level 4 — Problem Solving
3.4 Given: x² + (k − 2)x + 1 = 0 Prove that the roots of the equation are real for all values of k.

Source: DBE Mathematics Paper 1, November 2023, Question 3.4

See the progression — same type, increasing difficulty

Easy
Given: x² − 4x + 4 = 0 Determine, without solving the equation, the nature of the roots.

Practice question — not sourced from a past paper.

Medium
Given: x² + kx + 9 = 0 Determine the value(s) of k for which the roots of the equation are equal.

Practice question — not sourced from a past paper.

Hard
Given: x² − (k + 1)x + (k − 2) = 0 Prove that the roots of the equation are real for all real values of k.

Practice question — not sourced from a past paper.

Type 6: Word problems (disguised algebra)

Trigger words: No x is given — you must define the variable yourself

Trigger structure: "A rectangle has dimensions...", "Two numbers..." — context hides the equation.

Method (no numbers — just the steps)

  1. Define your variable first, in words
  2. Translate the context into an equation
  3. Solve the equation
  4. Check the answer makes sense in context (e.g. lengths can't be negative)
Grade 11Paper 1Level 3 — Complex Procedures
2.5 The length of a rectangle is 3 cm more than its width. The area of the rectangle is 40 cm². Determine the width of the rectangle.

Source: DBE Mathematics Paper 1, May/June 2023, Question 2.5

Common mistake

Writing an equation before defining what the variable represents.

See the progression — same type, increasing difficulty

Easy
TWO numbers differ by 3. The product of the two numbers is 40. Determine the two numbers.

Practice question — not sourced from a past paper.

Medium
The length of a rectangular garden is 4 m more than its width. The area of the garden is 60 m². Calculate the width of the garden.

Practice question — not sourced from a past paper.

Hard
A stone is thrown vertically upwards. Its height above the ground, h, in metres, after t seconds, is given by: h(t) = 20t − 5t² Determine the FIRST time, in seconds, at which the stone reaches a height of 15 m.

Practice question — not sourced from a past paper.

Words like determine and hence appear throughout this topic — see the instruction word glossary for full definitions.