Sometimes tested Papers 1 & 2

Algebra (Higher GCSE): Equations, Graphs, Sequences, Simultaneous Equations and Inequalities

Manipulating expressions and solving equations underpins almost every TMUA problem. This Higher-GCSE toolkit (factorising, the quadratic formula, curve shapes, sequences, simultaneous equations and inequalities) gets you to exact answers fast, without a calculator.

Part of the TMUA syllabus — revision for the Test of Mathematics for University Admission (TMUA), the UAT-UK maths test used by Cambridge, Oxford, Imperial, UCL, LSE, Warwick and Durham. No calculator; multiple choice.

Key points

  • Expanding and factorising are inverse operations; spotting a factorisation (difference of two squares a2 - b2 = (a - b)(a + b)) can collapse a hard-looking expression in one line.
  • A quadratic ax2 + bx + c = 0 can be solved by factorising, completing the square, or the formula (x = (-b ± √(b2 - 4ac))/(2a)); the discriminant b2 - 4ac tells you how many real roots exist before you solve.
  • Recognise curve shapes on sight: linear is a straight line, quadratic is a parabola, a cubic has up to two turning points, the reciprocal y = 1/x has two branches with asymptotes, an exponential y = a^x passes through (0, 1), and y = sin x and y = cos x oscillate between -1 and 1.
  • Arithmetic sequences add a fixed common difference d (n-th term a + (n-1)*d); geometric sequences multiply by a fixed common ratio r (n-th term a × rn-1).
  • For simultaneous equations use elimination when both are linear, and substitution when one is a curve; a line and a curve can meet at 0, 1 or 2 points.
  • Solve a linear inequality like an equation, but reverse the sign when you multiply or divide by a negative (e.g. -2x < 6 gives x > -3); read quadratic inequalities off a quick parabola sketch.
  • Completing the square rewrites ax2 + bx + c as a(x - h)2 + k, which exposes the vertex (h, k) and the minimum or maximum value directly.

Formulae

x = (-b ± √(b2 - 4ac))/(2a)

Solving a quadratic ax2 + bx + c = 0 that does not factorise neatly.

a2 - b2 = (a - b)(a + b)

Factorising or simplifying a difference of two squares in one step.

un = a + (n-1)*d

Finding the n-th term of an arithmetic sequence with first term a and common difference d.

un = a × rn-1

Finding the n-th term of a geometric sequence with first term a and common ratio r.

ax2 + bx + c = a(x + b/(2a))2 + c - b2/(4a)

Completing the square to locate a parabola's vertex and its minimum or maximum value.

D = b2 - 4ac

Counting the real roots of a quadratic (or intersections of a line and curve) without actually solving.

Definitions

Discriminant
The quantity b2 - 4ac from a quadratic ax2 + bx + c = 0. Positive gives two distinct real roots, zero gives one repeated root, and negative gives no real roots.
Root of an equation
A value of x that makes the equation true; graphically it is where the curve crosses the x-axis, i.e. where y = 0.
Asymptote
A line that a curve approaches ever more closely but never touches; for example y = 1/x approaches the x-axis and the y-axis.
y = 0
x = 0
Common difference / common ratio
The fixed number you add to get the next term of an arithmetic sequence (d), or the fixed number you multiply by in a geometric sequence (r).

Worked examples

1

The line y = 2x + 1 meets the curve y = x2 - x + 3. Find the coordinates of both points of intersection.

  1. 1

    Set curve equal to line:

    x2 - x + 3 = 2x + 1
  2. 2

    Move all terms to one side:

    x2 - 3x + 2 = 0.

  3. 3

    Factorise the quadratic:

    (x - 1)(x - 2) = 0.

  4. 4

    Read off the roots:

    x = 1 or x = 2
  5. 5
    Put x = 1 into the line:
    y = 3
  6. 6
    Put x = 2 into the line:
    y = 5

Answer: The line meets the curve at (1, 3) and (2, 5).

2

Solve the inequality x2 - x - 6 < 0, giving your answer as a range of values of x.

  1. 1

    Factorise the quadratic:

    (x - 3)(x + 2) < 0.

  2. 2

    Find the critical values:

    x = 3 and x = -2
  3. 3
    Sketch y = (x - 3)(x + 2):

    an upward parabola dipping below the x-axis between its roots.

  4. 4

    The expression is negative between the roots:

    -2 < x < 3.

Answer: -2 < x < 3.

Common mistakes

  • ×Forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number, so -2x < 6 is wrongly written as x < -3 instead of x > -3.
  • ×Sign slips when squaring a bracket: (a - b)2 = a2 - 2ab + b2, not a2 - b2.
  • ×Losing a solution by cancelling a factor of x; dividing x2 = 4x by x drops the root x = 0. Always collect terms on one side and factorise instead.
  • ×Confusing arithmetic and geometric sequences, for example checking a common difference when the terms are actually multiplied by a common ratio.
  • ×Stating a quadratic inequality straight from the roots without a sketch, and so giving the region outside the roots when the answer is between them (or vice versa).

No-calculator tips

  • ✓For a factorisable quadratic, find the factor pair of the constant term that adds to the middle coefficient by inspection; it is quicker than the formula.
  • ✓Use only the sign of the discriminant b2 - 4ac to answer 'how many roots' style options without solving anything.
  • ✓Sketch fast from intercepts, the sign of the leading coefficient and the general shape; the picture often settles the answer before any algebra.
  • ✓For 'how many times does the line meet the curve', substitute and test the discriminant of the resulting quadratic rather than finding the actual points.
  • ✓Substitute a convenient value such as x = 0 or x = 1 into candidate expressions or identities to eliminate wrong multiple-choice options quickly.

Test yourself

Original practice questions, no calculator. Work each out before revealing the answer.

Q1.What are the solutions of (x - 3)(x + 1) = 5?

  • A. x = 3 or x = -1
  • B. x = 4 or x = -2
  • C. x = -4 or x = 2
  • D. x = 4 only
Show answer

Answer: B — x = 4 or x = -2

The right-hand side is 5, not 0, so first expand: x2 - 2x - 3 = 5, giving x2 - 2x - 8 = 0 = (x - 4)(x + 2), so x = 4 or x = -2. Setting each bracket to 5's origin as if it were 0 gives the trap x = 3 or x = -1.

Q2.For which values of x is x2 < 5x - 6?

  • A. x < 2 or x > 3
  • B. 2 ≤ x ≤ 3
  • C. -3 < x < -2
  • D. 2 < x < 3
  • E. x < -3 or x > -2
Show answer

Answer: D — 2 < x < 3

Rearrange to x2 - 5x + 6 < 0, i.e. (x - 2)(x - 3) < 0, which holds strictly between the roots: 2 < x < 3. The region x < 2 or x > 3 is where the expression is positive (wrong condition).

Q3.A sequence begins 3, 8, 15, 24, 35, ⋯ What is its nth term?

  • A. 2n2
  • B. 5n - 2
  • C. n2 + 2n
  • D. n2 + 2
  • E. n2 + n
Show answer

Answer: C — n2 + 2n

The second difference is 2, so the n2 coefficient is 2/2 = 1; subtracting n2 (1, 4, 9, ⋯) leaves 2, 4, 6, ⋯ = 2n, giving n2 + 2n. Using the second difference 2 directly as the coefficient wrongly gives 2n2.

Read this topic in the official UAT-UK TMUA content specification →

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