Question

Why do you factor quadratic equations?

Answer 1

Because it tells you what the roots of the equation are, i.e. where `ax^2+bx+c=0`, which is often a useful thing to know.

Explanation:

Because it tells you what the roots of the equation are, i.e. where `ax^2+bx+c=0`, which is often a useful thing to know.

Think of it backwards - start by knowing that the quantity `x` is zero in two places, `A` and `B`. Then two equations describing `x` are `x-A=0` and `x-B=0`. Multiply them together:
`(x-A)(x-B)=0`
This is a factored quadratic equation.

Multiply out to get the unfactored equation:
`x^2-(A+B)x+AB=0`

So when you are presented with a quadratic equation, you know that the coefficient of the `x` term is the negative of the sum of the two roots and the constant coefficient is the product of them. This knowledge is usually a help in seeing if you can easily factor a quadratic. For example:
`x^2-11x+30=0`
Now we want two numbers that add to +11 and multiply to 30; the answers are 5 and 6, we see after trying a few, so it factors as `(x-5)(x-6)=0`.

Answer 2

By factorising first and then applying the multiplication property of zero, we can solve a quadratic equation.

Explanation:

One of the properties of `0` is that :

"Anything multiplied by `0` is equal to `0`"

So, if we have an equation where:

`a xx b xx cxx d xx e =0`,

then because of the multiplication property of `0`, we will know that at least one of the factors being multiplied must be equal to `0`.

Since we can't know which one is the `0`, we consider each in turn being `0`.

`:. a =0" or " b=0" or " c=0" "or" " d=0" "o r" " e=0`

However, this is only true for FACTORS.

So to apply this concept in solving a quadratic (or cubic, quartic, etc) equation, start by factorising to find the factors.

Then let each factor be equal to `0` and solve to find the possible values of the variable.

`x^2+5x=6" "larr` of no help in this form:

`x^2+5x-6=0" "larr` make it equal to `0`

`(x+6)(x-1)=0" "larr` two factors multiply to give `0`

Let each be equal to `0`

If `x+6=0" "rarr x =-6`

If `x-1=0" "rarr x =1`

By factorising first and then applying the multiplication property of zero, we can solve the quadratic equation.