Grade 12 · Mathematics
Quadratic equations
Solve quadratic equations by factorising or using the quadratic formula, and check every root.
Recognise the standard form
Write the equation as ax² + bx + c = 0, where a is not zero. Move every term to one side before choosing a method.
Choose a method
Factorise when the factors are easy to identify. Otherwise use x = (-b ± √(b² - 4ac)) / (2a). The discriminant b² - 4ac tells you whether real roots exist.
Formula sheet
- ax² + bx + c = 0
- x = (-b ± √(b² - 4ac)) / (2a)
- Discriminant Δ = b² - 4ac
Worked example
Solve x² - 5x + 6 = 0.
- Find two numbers whose product is 6 and whose sum is -5: -2 and -3.
- Factorise: (x - 2)(x - 3) = 0.
- Set each factor to zero: x = 2 or x = 3.
- Check: 2² - 5(2) + 6 = 0, and 3² - 5(3) + 6 = 0.
Try it yourself
1. Solve x² - 7x + 12 = 0. Check answer
(x - 3)(x - 4) = 0, so x = 3 or x = 4.
2. How many real roots does x² + 4x + 5 = 0 have? Check answer
Δ = 16 - 20 = -4. There are no real roots.
Common mistake: Do not divide by x while solving: it can remove the valid root x = 0.