0580

Quadratic Equations

Algebra and Sequences

The standard shape

  • A quadratic equation contains a squared term and nothing of higher power, and its standard shape is ax² + bx + c = 0, where a is not zero
  • All three methods below need zero on one side, so rearranging comes before choosing a method
  • Collect everything on the side that leaves the x² term positive; if the squared term ends up negative, multiply every term by −1
    • 12 − x² = 4x becomes 0 = x² + 4x − 12, which is easier to work with than −x² − 4x + 12 = 0
  • The rearranged equation has three terms, and reaching it correctly is worth a mark on its own even before any solving happens
  • Most quadratics give two solutions; some give one repeated value, and a few have none

Which method to use

  • Factorising is quickest whenever the expression factorises with whole numbers
  • The formula always works, and is the method to reach for when the numbers are awkward or the answer is wanted as a decimal
  • Completing the square always works too, and is the method asked for when the question wants a squared bracket or a turning point
  • A question that says "solve" without naming a method leaves the choice to you, so a failed attempt at factorising costs only time, not marks
Exam tip

Rearrange to = 0 before choosing a method. None of the three routes works on an equation with terms on both sides, and the rearranging itself is usually the first mark. Collect on the side that leaves x² positive: it saves a sign error later.

Worked example

Getting an equation into the standard shape

Solve 2x² = 5x + 12.

Solution:

  • Nothing can be done until every term is on one side and the other side is zero
  • Subtract 5x and 12 from both sides: 2x² − 5x − 12 = 0
  • Three terms with a number in front of x², so try factorising first
  • Find two numbers multiplying to 2 × (−12) = −24 and adding to −5: they are −8 and 3
  • Split the middle term: 2x² − 8x + 3x − 12 = 0
  • Group: 2x(x − 4) + 3(x − 4) = 0, so (2x + 3)(x − 4) = 0
  • A product is zero only if a bracket is zero, so 2x + 3 = 0 or x − 4 = 0
  • x = −1.5 or x = 4
  • Check the first: 2 × (−1.5)² = 4.5, and 5 × (−1.5) + 12 = 4.5