Quadratic equations

Last time

Before we start: last time, in Algebraic fractions, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Two answers

A ball is thrown up and passes the same height twice: once on the way up and once on the way down. That is why a quadratic equation usually has two solutions. Today you solve them three ways: by factorising, with the formula, and by completing the square.

The rule that makes it work

Here is the key fact. If two things multiply to zero, at least one of them must be zero. So if x minus seven times x plus two equals zero, then either x minus seven is zero, or x plus two is zero. That gives x equals seven, or x equals minus two. Two answers. The equation must equal zero first. If it does not, rearrange it.

Solving by factorising

Step one: rearrange to make one side zero. Step two: factorise the other side. Take x squared minus five x minus fourteen equals zero. Two numbers that multiply to minus fourteen and add to minus five: minus seven and two. So x minus seven, times x plus two, equals zero. Step three: set each bracket to zero. Step four: x is seven, or x is minus two.