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Algebra and Functions

Simultaneous Equations and Inequalities

Pearson Edexcel A-level Mathematics

Linear and quadratic simultaneous equations

Method: substitution
  1. Rearrange the linear equation to make one variable the subject.
  2. Substitute into the quadratic equation.
  3. Collect into a three-term quadratic equal to zero and solve it.
  4. Substitute each root into the linear equation to find its paired value.
  5. Write each solution as a pair of coordinates.
Worked example: points where a line meets a curve

The line l has equation 3x − y = 1. The curve C has equation y = 2x2 − 4x + 5. Use algebra to find the coordinates of the points where l meets C.

y = 3x − 1

2x2 − 4x + 5 = 3x − 1 ⇒ 2x2 − 7x + 6 = 0

(2x − 3)(x − 2) = 0 ⇒ x = 32, x = 2

y = 3 × 32 − 1 = 72,   y = 3 × 2 − 1 = 5

(32, 72) and (2, 5)

Quadratic inequalities

Method: solving a quadratic inequality
  1. Rearrange to a quadratic compared with zero.
  2. Solve the quadratic equal to zero for the critical values.
  3. Sketch the curve through the critical values.
  4. Choose the region, with a positive x2 coefficient: outside the critical values for > 0, between them for < 0.
x a b
A quadratic with a positive x2 coefficient and critical values a and b: f(x) > 0 when x < a or x > b, and f(x) < 0 when a < x < b.
  • Multiplying or dividing by a negative number reverses the inequality sign.
  • For an inequality with x in a denominator, multiply both sides by the square of the denominator, never by an expression whose sign is unknown.
Worked example: quadratic inequality in set notation

Find the set of values of x for which x(2x + 7) > 4, giving your answer in set notation.

2x2 + 7x − 4 > 0

(2x − 1)(x + 4) = 0 ⇒ critical values x = −4, x = 12

Outside region: x < −4 or x > 12

{x : x < −4} ∪ {x : x > 12}

Set notation

RegionInequalitySet notation
Outsidex < a or x > b{x : x < a} ∪ {x : x > b}
Betweena < x < b{x : a < x < b}
  • Use or and ∪ for two separate intervals; use and and ∩ for one interval: {x : x > a} ∩ {x : x < b} is a < x < b.
  • When set notation is asked for, an inequality without the set brackets is not a complete answer.

Regions

A strict inequality, < or >, is drawn with a dotted boundary, and ≤ or ≥ with a solid boundary. A region between a curve and a line is one double inequality, f(x) < y < g(x).

These notes are part of an Excel with Osman subscription and are not available to print.

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