Quadratic Equations
Factorising, the quadratic formula, completing the square, and quadratic graphs.
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Every point below, worked through in one video, and each point links straight to its own part.
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Quadratic Equations, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Solve quadratics of the form x² + bx + c = 0 by factorising
Factorise, then use the fact that if two things multiply to zero, one of them IS zero. x² + 5x + 6 = 0 factorises to (x + 2)(x + 3) = 0, so x = −2 or x = −3. Always rearrange to "= 0" before factorising - that fact only works against zero.
Read the roots of a quadratic from its graph
The roots are where the curve crosses the x-axis - the x values that make y zero. A quadratic can cross twice, touch once (a repeated root), or miss entirely (no real roots). The turning point sits exactly halfway between the two roots.
Solve quadratics where the x² coefficient is more than 1Higher only
For 2x² + 7x + 3 = 0, find two numbers that multiply to 2 × 3 = 6 and add to 7 (they are 6 and 1), split the middle term and factorise in pairs: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). So x = or x = −3.
Solve quadratics using the quadratic formulaHigher only
When factorising fails, use x = with the equation in the form ax² + bx + c = 0. Write down a, b and c first, brackets around negatives - most formula errors are sign errors. The ± gives the two solutions; round only at the very end.
Complete the square, and use it to solve equations and find turning pointsHigher only
x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7: halve the x coefficient, then subtract its square. This form solves equations exactly - (x + 3)² = 7 gives x = −3 ± - and hands you the turning point, here (−3, −7), for free.
Set up quadratic equations from area and other worded problems
When the unknown multiplies itself - a rectangle x by (x + 3) with area 40 - the equation is quadratic: x² + 3x − 40 = 0. Solve as usual, then sense-check both roots against the context: a length cannot be negative, so discard the root that does not fit.
Common mistakes in Quadratic Equations
The slips that cost marks year after year, and what to do instead.
Factorising before rearranging to = 0, so x² + 3x = 10 turns into x(x + 3) = 10.
Get everything on one side first: x² + 3x − 10 = 0 factorises to (x + 5)(x − 2) = 0, so x = −5 or x = 2.
Copying the signs from the brackets, so (x + 2)(x + 3) = 0 gives x = 2 and x = 3.
Set each bracket equal to zero and solve: x + 2 = 0 gives x = −2, and x + 3 = 0 gives x = −3.
Squaring a negative on the calculator without brackets, so −3² gives −9 in a table of values for y = x².
Type (−3)² with the brackets: it is 9. The y values of y = x² mirror each other either side of x = 0, so −3 and 3 both give 9.
Leaving out the brackets when b is negative in the quadratic formula, so b² becomes −16 for b = −4.Higher only
Put negative values in brackets before you substitute: for x² − 4x + 3 = 0, b² − 4ac = (−4)² − 4 × 1 × 3 = 16 − 12 = 4.
Forgetting to take off the square when completing the square, so x² + 6x + 2 becomes (x + 3)² + 2.Higher only
Halving the 6 gives (x + 3)², which brings an extra 9, so take it off: x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7.
Practice
Try a Quadratic Equations question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
Answer: x = 4 or x = −3
Factorising gives (x − 4)(x + 3) = 0, because −4 and +3 multiply to −12 and add to −1.
- Find two numbers with product −12 and sum −1: they are −4 and +3.
- Factorise: x² − x − 12 = (x − 4)(x + 3).
- Set each bracket to zero: x − 4 = 0 or x + 3 = 0.
- So x = 4 or x = −3.
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