Quadratic Equations

Students learn how to  solve a range of quadratic equations using factorisation, applying the formula and completing the square.  Learning progresses from solving equations to deriving and sketching quadratics as a graph.

This unit takes place in  Year 11 Term 1 and follows on from solving linear and simultaneous equations.


Quadratic Equations Lessons
Problem Solving and Revision Lessons
Prerequisite Knowledge
  • Simplify and manipulate algebraic expressions by:
  • Expanding products of two or more binomials
  • Factorising quadratic expressions of the form x2 + bx + c, including the difference of two squares
  • Simplifying expressions involving sums, products and powers, including the laws of indices
  • Factorising quadratic expressions of the form ax2 + bx + c.
Success Criteria
  • Know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs
  • Simplify and manipulate algebraic expressions by factorising quadratic expressions of the form ax2 + bx + c.
  • Understand and use standard mathematical formulae; rearrange formulae to change the subject
  • Identify and interpret roots, intercepts, turning points of quadratic functions graphically
  • Deduce roots algebraically and turning points by completing the square
  • Recognise, sketch and interpret graphs of quadratic functions
  • Solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph
  • Solve two simultaneous equations in two variables linear/quadratic algebraically; find approximate solutions using a graph
Key Concepts
  • Check brackets have been factorised correctly by multiplying them back out.
  • To solve quadratics by factorising students need to identify two numbers that have a product of c and a sum of b. Roots are found when each bracket is made to equal zero and are solved for x.
  • When a quadratic cannot be solved by factorising students should use completing the square or the quadratic formula.
  • Students should be able to derive the quadratic formula from the method of completing the square.
  • A sketched graph is drawn freehand and includes the roots, turning point and intercept values.
  • Quadratic identities in the form (x + a)2 + b ≡ ax2 + bx + c can be solved either through completing the square to RHS = LHS or by expanding the brackets to LHS = RHS and equating the unknowns.
  • Quadratic and linear simultaneous equations should be sketched before solved algebraically to ensure students know to find and the x and y values.
Common Misconceptions
  • The method of trial and improvement is often incorrectly used to try and solve quadratics.
  • When solving quadratic and linear simultaneous equations students often forget to find the y values as well the x.
  • When using the formula to solve quadratics students often forget to evaluate the negative solution. Some students also incorrectly apply the division by reducing the terms it covers.
  • Students tend to struggle deriving quadratic equations from geometrical facts.

Go ad-free and get access to over 500 lessons

Mr Mathematics Blog

Two-Way Tables and Frequency Trees

Problem solving lesson on two-way tables and frequency trees.

Plotting Curved Graphs

Three typical exam questions to revise on plotting quadratic, cubic and reciprocal graphs.

Interpreting Cumulative Frequency Graphs

Linking cumulative frequency graphs to ratio, percentages and financial mathematics.