MathematicsExam code: 0580
Quadratic Equations
Algebra and Sequences
Why a zero product helps
- If two quantities multiply to give zero then at least one of them must be zero, and this is the whole idea behind the method
- Factorise the left-hand side into two brackets, then set each bracket equal to zero in turn
- Each of those small equations is linear and gives one solution
- Where the brackets are (x + m) and (x + n), the solutions are simply −m and −n, but that shortcut fails as soon as a bracket has a number in front of x
Worked example
Two brackets with numbers in front of x
Solve 4x² − 4x − 15 = 0.
Solution:
- The equation already has zero on one side, so factorise straight away
- Two numbers multiplying to 4 × (−15) = −60 and adding to −4 are −10 and 6
- Split the middle term: 4x² − 10x + 6x − 15 = 0
- Group in pairs: 2x(2x − 5) + 3(2x − 5) = 0
- Factorise: (2x − 5)(2x + 3) = 0
- Set each bracket to zero: 2x − 5 = 0 gives x = 2.5, and 2x + 3 = 0 gives x = −1.5
- The solutions are x = 2.5 or x = −1.5
Exam tip
Give both roots only when two are asked for
When a question asks you to solve and prints two answer spaces, give both roots, even if x is a speed and one root is negative: the answer mark needs the pair. When it asks for one quantity, such as a length, a radius or a speed, and prints one space, give only the root that makes sense. The last mark is for that value alone, and two schemes withheld it from an answer giving both roots.
When x itself is a factor
- An equation with no constant term, such as 3x² = 7x, factorises to x(3x − 7) = 0
- Treat the lone x as a bracket of its own, so it supplies the solution x = 0
- Never divide both sides by x to tidy the equation up, because dividing by something that might be zero throws that solution away
- Losing x = 0 in this way leaves only one of the two solutions, so the answer is incomplete