Quadratic equations: factorising, completing the square, the formula
Solve quadratic equations by factorising, complete the square with tiles as al-Khwarizmi did, and use the discriminant to see how many solutions there are.
- UK: Year 10
- France: grade 9
- Germany: grade 9
- Iran: grade 10
- Afghanistan: grade 9
- Syria: grade 9
- Palestine: grade 8
- Ukraine: grade 8
- Russia: grade 8
Guide
This studio follows GCSE A18 (Higher): solving quadratic equations by factorising, by completing the square and with the quadratic formula.
If a product is 0, one of its factors is 0. So (x − 2)(x − 3) = 0 gives x = 2 or x = 3. A quadratic equation can have two root (of an equation)s: type both.
Never divide both sides by x: x could be 0, and that solution would be lost. Move everything to one side and factorise.
completing the square: x² + 6x is a square x by x and two strips of 3x, one along each side. The missing corner is 3 × 3 = 9, so x² + 6x + 9 = (x + 3)². Muhammad ibn Musa al-Khwarizmi solved equations with this picture about 1200 years ago.
Doing the same with ax² + bx + c = 0 gives the quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac says how many real roots: two if b² − 4ac > 0, one if b² − 4ac = 0, none if b² − 4ac < 0.
On the factor board, tap the chip and Split, then type the roots in the boxes x = □. In the rectangle, build the two sides with − and +.