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.

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