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Grade 12 Algebra Equations Inequalities — How to Identify the Question Type

Grade 12 Algebra builds on Grade 11, but the DBE examination guidelines change which techniques are actually examinable. Completing the square is no longer used as a solving method — instead, two new types appear: the substitution (k-) method and surd equations. Inequalities and nature of roots also shift to be read in the context of functions and graphs rather than algebra alone.

Type 1: Solving a quadratic equation

Trigger words: "Solve for x", "Find the roots"

Trigger structure: A genuine quadratic in x — no repeated sub-expression, no surd.

Do not confuse with: The substitution method (Type 2) — only needed when the equation isn't quadratic in x itself.

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 12Paper 1Level 2 — Routine Procedures
1.1 Solve for x: 3x² + 5x − 2 = 0

Source: DBE Mathematics Paper 1, November 2022, Question 1.1

Common mistake

Reaching for the substitution method on an equation that's already a plain quadratic in x.

See the progression — same type, increasing difficulty

Easy
Solve for x: x² − 7x + 10 = 0

Practice question — not sourced from a past paper.

Medium
Solve for x: 2x² = 5x + 3

Practice question — not sourced from a past paper.

Hard
Solve for x (correct to TWO decimal places): 3x² − 4x − 5 = 0

Practice question — not sourced from a past paper.

Type 2: Solving using the substitution (k-) method

Trigger words: An equation that is quadratic in some repeated expression, not in x directly

Trigger structure: Look for the same expression appearing twice at different powers — e.g. x⁴ and x², or 2ˣ and 2²ˣ, or a repeated bracket like (x² − x).

Do not confuse with: A plain quadratic (Type 1) — if the equation is already quadratic in x, substitution is unnecessary extra work.

Method (no numbers — just the steps)

  1. Spot the repeated expression and let k equal it
  2. Rewrite the equation as a quadratic in k
  3. Solve the quadratic for k
  4. Substitute back and solve for x using each valid value of k
  5. Reject any k-value that's impossible for the original expression (e.g. k < 0 when k = 2ˣ)
Grade 12Paper 1Level 3 — Complex Procedures
1.3 Solve for x: x⁴ − 5x² + 4 = 0

Source: DBE Mathematics Paper 1, May/June 2022, Question 1.3

Common mistake

Solving for k and stopping there — forgetting to substitute back and solve for x.

See the progression — same type, increasing difficulty

Easy
Solve for x: x⁴ − 5x² + 4 = 0

Practice question — not sourced from a past paper.

Medium
Solve for x: 2^(2x) − 2^(x+1) − 8 = 0

Practice question — not sourced from a past paper.

Hard
Solve for x: (x² − x)² − 8(x² − x) + 12 = 0

Practice question — not sourced from a past paper.

Type 3: Equations involving surds

Trigger words: A square root sign in the equation

Trigger structure: Isolating the surd and squaring both sides leads to a quadratic equation.

Do not confuse with: Simplifying surds — this type asks you to solve for x, not just simplify an expression.

Method (no numbers — just the steps)

  1. Isolate the surd term on one side of the equation
  2. Square both sides
  3. Solve the resulting quadratic equation
  4. Substitute every solution back into the ORIGINAL (un-squared) equation
  5. Reject any solution that doesn't satisfy the original equation
Grade 12Paper 1Level 3 — Complex Procedures
1.4 Solve for x: √(x + 6) = x

Source: DBE Mathematics Paper 1, November 2021, Question 1.4

Common mistake

Accepting both roots of the quadratic without checking them in the original equation — squaring can introduce a root that doesn't actually work.

See the progression — same type, increasing difficulty

Easy
Solve for x: √(x + 6) = x

Practice question — not sourced from a past paper.

Medium
Solve for x: √(2x + 3) = x

Practice question — not sourced from a past paper.

Hard
Solve for x: x − √(x + 2) = 4

Practice question — not sourced from a past paper.

Type 4: Simultaneous equations

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

Trigger structure: Two equations, two unknowns — substitute the linear equation into the non-linear one.

Method (no numbers — just the steps)

  1. Make one variable the subject of the linear equation
  2. Substitute into the second equation
  3. Solve the resulting equation
  4. Substitute back to find the other variable
Grade 12Paper 1Level 3 — Complex Procedures
2.2 Solve the following equations simultaneously for x and y: y = x + 1 x² + 2xy − y² = 8

Source: DBE Mathematics Paper 1, November 2022, Question 2.2

See the progression — same type, increasing difficulty

Easy
Solve simultaneously for x and y: y = x − 3 y = x² − 5x + 3

Practice question — not sourced from a past paper.

Medium
Solve simultaneously for x and y: 2x − y = 4 x² + y² = 20

Practice question — not sourced from a past paper.

Hard
Solve simultaneously for x and y: y = x + 2 x² + 3xy − y² = 4

Practice question — not sourced from a past paper.

Type 5: Inequalities in the context of functions

Trigger words: "Use the graph to determine...", "for which values of x is f(x) ≥ 0"

Trigger structure: A graph (often cubic or rational) is given, and you must read the inequality from its shape rather than solving algebraically.

Do not confuse with: Pure algebraic inequalities — at Grade 12, non-quadratic inequalities are read from a graph, not factorised from scratch.

Method (no numbers — just the steps)

  1. Identify the x-intercepts (and any asymptotes) shown on the graph
  2. Note where the graph is above or below the x-axis between those intercepts
  3. Match the question's inequality symbol to the correct intervals
  4. Write the answer in the required notation, excluding any x-value where the function is undefined
Grade 12Paper 1Level 3 — Complex Procedures
6.4 The diagram shows the graph of f(x) = (x + 1)(x − 2)(x − 4). Use the graph to determine the values of x for which f(x) ≤ 0.

Source: DBE Mathematics Paper 1, May/June 2023, Question 6.4

See the progression — same type, increasing difficulty

Easy
The graph of g(x) = (x + 2)(x − 3) is shown, cutting the x-axis at x = −2 and x = 3. Use the graph to determine the values of x for which g(x) > 0.

Practice question — not sourced from a past paper.

Medium
The graph of h(x) = (x + 1)(x − 1)(x − 3) is shown. Determine the values of x for which h(x) ≥ 0.

Practice question — not sourced from a past paper.

Hard
The graph of k(x) = (x − 2)/(x + 1) is shown, with a vertical asymptote at x = −1. Determine the values of x for which k(x) < 0.

Practice question — not sourced from a past paper.

Type 6: Nature of roots, read from a function or graph

Trigger words: "For which value(s) of k does the graph touch the x-axis", "...does not intersect the x-axis", "...has two x-intercepts"

Trigger structure: The discriminant idea from Grade 11, but phrased through a graph: equal roots = touches the x-axis, real unequal roots = two x-intercepts, non-real roots = no x-intercepts.

Method (no numbers — just the steps)

  1. Translate the graph description into a discriminant condition (touches = Δ = 0, two intercepts = Δ > 0, no intercepts = Δ < 0)
  2. Identify a, b and c from the function's equation
  3. Set up and solve the resulting equation or inequality in k
Grade 12Paper 1Level 3 — Complex Procedures
5.3 Given: f(x) = x² + kx + 9 Determine the value(s) of k for which the graph of f touches the x-axis.

Source: DBE Mathematics Paper 1, November 2023, Question 5.3

Common mistake

Trying to read the answer off a sketch by eye instead of setting up the discriminant condition algebraically.

See the progression — same type, increasing difficulty

Easy
Given: f(x) = x² − 6x + k Determine the value of k for which the graph of f touches the x-axis.

Practice question — not sourced from a past paper.

Medium
Given: g(x) = x² + 2x + (k − 3) Determine the value(s) of k for which the graph of g does not intersect the x-axis.

Practice question — not sourced from a past paper.

Hard
Given: h(x) = (k − 1)x² − 4x + 1 Determine the value(s) of k for which the graph of h has two distinct x-intercepts.

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.