Solving quadratic equations using the zero product method

Hi Kim! See the twelve there? Do you know how to spit it up into several factors? Right. You've learned that. 'Twelve' equals 'three' times 'four'.

And because 'four' equals 'two-times-two' we can write it out like this: 'Twelve' equals 'three' times 'two' times 'two'. What we just did is called factorisation, or factoring. We took the number twelve and wrote it out as a product of three factors. You can factor, or factorise all composite numbers (in other words those which aren't prime numbers) but you can also factorise mathematical expressions. For example, look at the expression 'x-squared' plus 'three-x'.

We have two terms in the expression, and both terms have the factor 'x' in common. So we can take out 'x' and write the expression like this: 'x' times 'x-plus-three' - in brackets. We have written the expression as a product of the factors 'x' and 'x-plus-three'. We have performed a factorisation of the expression. Let's take another example.

Can we factorise the expression 'two-x-squared' plus '8-x' plus '8'? All the terms have the factor 'two' in common. So we can take out 'two', and the expression equals 'two' - open brackets - 'x-squared' plus 'four-x' plus 'four' - close brackets. Now look at the expression in brackets. Does the pattern remind you of anything?

Remember the first rule for squaring a binomial? There's another film about that. The expression we see here has the same pattern we see when we expand a binomial using the first rule for squaring a binomial. So let's try it, working backwards. 'x-squared' plus 'four-x' plus 'four' equals (x-plus-two) to the power of 'two'. This means that our first expression can be written 'two times x' plus 'two-to-the-power-of-two', and we have done a factorisation in multiple steps.

What about this expression? Can it also be factorised? Here we can use the second rule for squaring a binomial, and we see that the expression equals x-minus-four to the power-of-two. How can we apply this? We can solve quadratic equations using factorisation.

Go back to the first expression, 'x-squared' plus 'three-x'. We make the expression the left-hand side of an equation and the right-hand side zero - like so. How do we solve the equation? We factorise, like before. In this case we take out x, and so we can write that: 'x' times 'x-plus-three' equals zero.

Anything multiplied by zero always equals: zero. This means that if any of the factors is zero, the whole expression is zero. And this is how we find the equation's solution. If x equals zero, the entire expression is zero. Therefore 'x equals zero' is one of the equation's solutions.

For the next solution, we get 'x plus three' equals 'zero'. This means that 'x' equals 'minus-three' is the second solution of the equation. One more example, using another expression we saw recently. If we say: 'Two x squared' Plus 'eight x' Plus 'eight' Equals 'zero', How do we solve this equation? When we factorise it, the left side becomes: 'Two', times 'X plus two' 'To the power of two' And it also equals 'zero'.

We can divide both sides by two. Take the 'two' off there; And 'zero' divided by 'two' is still: 'zero'. That means that 'x plus two' 'to the power of two' also equals: 'zero'. We're near to the equation's solution as we write: 'x plus two' equals 'zero'. x equals minus two.

Thus, we can solve some quadratic equations by factorising the expressions. Begin with rewriting the equation so that the right side is zero. See if you can factorize the left hand side. Look for patterns like: a common denominator, a difference of squares, and the two rules of squaring a binomial. If we can factorise, we can find the equation's solution if one of the factors can be set as 'zero'.

Solve the Equation! So that's factorisation!