GCSE Quadratics: Factorise, Complete the Square or Use the Formula?
Three ways to solve a quadratic, and students learn them as three separate topics. Here is how each one works and how to choose in the exam.
Most GCSE students learn three ways to solve a quadratic equation, usually weeks apart, and file them as three separate topics. Then an exam question gives them x² + 6x + 5 = 0 and they freeze, because nobody told them which of the three to use.
The answer is that all three work, and the question usually tells you which one it wants. This post shows each method on the same equation, then gives a simple rule for choosing.
Route 1, factorising
Factorising turns the quadratic into two brackets that multiply to zero. For x² + 6x + 5, your child looks for two numbers that multiply to 5 and add to 6. Those are 1 and 5, so the equation becomes (x + 1)(x + 5) = 0.
If two things multiply to make zero, one of them must be zero. So x + 1 = 0 or x + 5 = 0, which gives x = −1 or x = −5.
When there is a number in front of x², I teach one method that always works. Take 2x² + 7x + 3. Multiply the 2 and the 3 to get 6. Find two numbers that multiply to 6 and add to 7, which are 1 and 6. Write (2x + 1)(2x + 6), then take out anything that divides a whole bracket, here the 2 from the second one, to get (2x + 1)(x + 3). Expand it to check, and the solutions are x = −½ or x = −3.
One special case is worth knowing on sight. x² − 49 has no middle term, and it factorises straight to (x + 7)(x − 7), because 49 is a square number. This is called the difference of two squares.
The factorising quadratics skill in our free Maths Skills Lab has nine levels, from all positive numbers up to a number in front of x² and the difference of two squares, and its Teach mode shows each step one press at a time.
Route 2, completing the square
Completing the square rewrites the quadratic as a squared bracket plus or minus a number. Halve the 6 to get 3, write (x + 3)², then correct for the extra 9 that bracket creates. So x² + 6x + 5 becomes (x + 3)² − 9 + 5, which is (x + 3)² − 4.
To solve, set it equal to zero. (x + 3)² = 4, so x + 3 is 2 or −2, and x = −1 or x = −5, the same answers as before. Remembering the negative square root is where most students drop a mark.
This route earns its place in two kinds of question. When the question says "write in the form (x + a)² + b", it wants this method by name. And the completed square gives the turning point of the graph for free. For x² + 8x + 10 = (x + 4)² − 6, the lowest point of the curve is at (−4, −6).
The completing the square skill has ten levels, including a number in front of x², finding the turning point and rearranging before you solve.
Route 3, the quadratic formula
The formula always works, including when the answers are not whole numbers. Your child reads off a, b and c, works out b² − 4ac first, then puts everything into x = (−b ± √(b² − 4ac)) ÷ 2a.
For 2x² − 3x − 4 = 0, a = 2, b = −3 and c = −4. Then b² − 4ac is 9 + 32, which is 41, and x = (3 ± √41) ÷ 4. That gives x = 2.35 or x = −0.85 to two decimal places.
Three slips cost most of the marks here. When b is negative, −b is positive, so −(−3) is 3. When b is negative, b² is still positive, so (−3)² is 9. And the whole top line is divided by 2a, not only the square root. Working out b² − 4ac on its own first avoids all three.
The quadratic formula skill climbs through seven levels, from finding a, b and c to rounded answers and rearranging first.
How to choose in the exam
- Read the question for a clue. "Factorise" means factorise. "In the form (x + a)² + b" means complete the square. "Give your answers to 2 decimal places" or "to 3 significant figures" means the formula, because the answers will not be whole numbers.
- With no clue, try factorising first. If a pair of numbers does not appear within a minute, stop looking and use the formula. It works every time.
- Use completing the square for turning points. Any question about the lowest or highest point of a quadratic graph is a completing the square question.
A useful check is b² − 4ac. If it is a square number, such as 16 or 49, the quadratic factorises. If it is not, go straight to the formula. If it is negative, there are no solutions at all.
Where quadratics turn up on the Higher paper
Quadratics rarely appear alone near the end of a Higher paper. They sit inside inequalities, simultaneous equations, algebraic fractions and graphs. The last question on the 2023 Edexcel Higher Paper 1 asks for the values that satisfy two quadratic inequalities at once, and it needs factorising twice before the real question even starts.
Our step by step walkthrough of that question is free in the Skills Lab, alongside the other hardest questions from two 2023 Higher papers.
A ten minute routine
Once a week, give your child one quadratic and ask them to solve it all three ways. It takes about ten minutes, and getting the same two answers three times is the fastest way to see that the methods are connected rather than three things to memorise.
For more help, our GCSE maths tutors teach this one to one, and our mock marking service shows exactly which questions on a real paper are losing marks.
Aadam, SHLC Tutors
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