IGCSE Maths / Quadratic equations, curves and inequalities
The quadratic formula
Edexcel International GCSE Maths (4MA1), chapter 22: Quadratic equations, curves and inequalities, specification point 2.7b. Free lesson.
Read the revision notes · IGCSE Maths past papers
The quadratic formula solves any quadratic equation, including ones that do not factorise. Read the coefficients carefully, substitute them with brackets around negatives, and use both signs to find both answers.
What the quadratic formula uses
A quadratic equation contains an x² term. The formula is printed on the Edexcel formula sheet, so the key skill is identifying the values to substitute. First arrange the equation as ax² + bx + c = 0. The signs belong to the coefficients, so include them when you read off a, b and c. a is the number in front of x². b is the number in front of x. c is the number on its own at the end.
How to substitute and solve
Put the values into the formula, using brackets around every negative number. This makes the intended calculation clear, especially when a negative value is squared. Work out the expression under the square root first. Then calculate one answer using + and the other using −. The expression b² − 4ac is under the square root. The ± sign means there are usually two answers. Show the substitution line, even if you use a calculator for the later steps. It can earn a mark on its own.
Worked example: 3x² − 7x + 1 = 0
This quadratic does not factorise, so use the formula. Here a = 3, b = −7 and c = 1. Keep the brackets around −7 when substituting.
Choosing a method and avoiding a sign error
Factorising is quickest when whole-number brackets work. Completing the square and the quadratic formula work for all quadratics. Factorising gives whole numbers or simple fractions, while the other two methods can give surds or decimals. The formula is printed on the sheet; the other methods are not. Try factorising briefly, but move to the formula if no whole-number pair works after about twenty seconds. Any of the three methods earns full marks unless the question names a method. A request to solve by factorising, or to give answers to 2 decimal places, indicates the expected method.…
Quick check and final reminders
For 2x² + 5x − 1 = 0, a = 2, b = 5 and c = −1. The negative value of c must keep its sign in the expression under the root. Substitution gives x = (−5 ± √33)/4. Rearrange to equal zero first, then identify a, b and c with their signs. Use brackets for every negative, work out under the root first, and use both the plus and minus answers.