Quadratic equations
Factorising, the discriminant, the nature of roots and word problems.
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A quadratic has an x². Its graph is a U shape, and it has up to two answers.
Look: Blue crosses twice (2 answers). Green just touches (1 answer). Red misses (no real answer).
✏️ Your turn!
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1Solve x² − 7x + 12 = 0
2Solve x² = 25
3Solve x² = 5x
! Careful!
✗Dividing x² = 5x by x.
✓You lose x = 0. Factorise: x(x − 5) = 0.
✗Thinking every quadratic has answers.
✓If the U misses the x-axis, there are no real answers.
See the whole lesson on one page
A quadratic has an x². Its graph is a U shape, and it has up to two answers.
To solve, factorise if you can. If not, use the formula. Some U-shapes cross the x-axis twice, some just touch it, some miss it.
Think of it like thisA ball hit high goes up and comes down in a curve. That curve is a quadratic.
- Root
- An answer of the equation.
Solved together
Two numbers next to each other multiply to 306. Find them.
Call them x and x + 1.
x(x + 1) = 306x² + x − 306 = 0Two numbers that multiply to −306 and add to 1: 18 and −17.
(x + 18)(x − 17) = 0We want positive numbers, so x = 17.
17 × 18 = 306 ✓
Answer17 and 18.
For older learners and grown-ups: a little more
Formula: x = [−b ± √(b² − 4ac)] / 2a. The part D = b² − 4ac decides: D > 0 two roots, D = 0 one, D < 0 none. Example: 2x² − 4x − 3 = 0 gives x = (2 ± √10)/2.
Now practise it
Kanakku makes new questions on quadratic equations every time. It shows the working and gives a hint when you are stuck.