Factorising quadratic expressions, and solving quadratic equations

What do we mean by factorising?

Factorising - or factoring, to non-British English speakers - is the process of breaking an expression into its factors.

You may have seen how to factorise a linear expression by taking out a common factor:

2x + 4 = 2(x + 2) as 2 is a common factor of both 2x and 4.

This process is the opposite of expanding the bracket - and the same is true for quadratics.

What does "quadratic" mean?

A quadratic is an algebraic expression where the highest power of the unknown variable is 2 - in other words, anything in the form ax2 + bx + c.  Generally, we write them in descending order; the x2 term first, then the x term, then the constant c.

Factorising as the inverse of expanding

When you worked on expanding a double bracket (x + a)(x + b), you may have noticed that:

  • The numbers in the brackets always add up to the number of "xs," or the "coefficient of x."
  • The numbers in the brackets always multiply to make the +? on the end, or "the constant term."

If we expand (x + a)(x + b) we get x2 + ax + bx + ab, or x2 + (a+b)x + ab - which supports the observations above...

So, to factorise a quadratic we need to find two numbers that add up to the x coefficient, and multiply to make the constant on the end. For example:

"Factorise x2 + 7x + 12"

We need two numbers that:

  • Sum to 7
  • Have a product of 12

From this we can (hopefully) see that the numbers we need are 3 and 4.

Top tip: Look for factors of the constant first, as there are a finite number of possibilities! In this case, the only possible integer pairs are: 1 & 12, 2 & 6, 3 & 4, -3 & -4, -2 & -6, -1 & -12. Then find the pair that sum to the number you need.

So how do I solve quadratic equations?

The "solutions," or roots, of a quadratic are the values of x where the graph would cross - or touch - the x axis. In other words, they are the x values that make the whole equation equal zero.

Because we are multiplying our two linear factors (the pair of brackets) together to make our quadratic, we can make the whole expression zero by making either bracket zero; anything multiplied by zero is still zero!

So to solve x2 + 7x + 12 = 0:

  1. Factorise: (x + 3)(x + 4)
  2. What x value makes the first bracket zero? -3 + 3 = 0, so x = -3.
  3. What x value makes the second bracket zero? -4 + 4 = 0, so x = -4.

This means our roots are x = -3 and x = -4.  If you plotted the graph of y = x2 + 7x + 12, it would cross the x-axis at these points.

If you have a quadratic that won't factorise, you need to learn the Quadratic Formula - but it's not too hard!

For a quadratic a(x squared) + bx + c:

Take the factorising quiz:

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2 comments

Babbler87 profile image

Babbler87 5 years ago from Ho Chi Minh City (Saigon), Viet Nam

Funniest video on quadratics!


Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man

Brilliant1 This is a great treatment of solving quadratics and the video adds an element of fun that would definitely make it stick in any student's long term memory. I will be visiting regularly as result of reading this hub. Thank you.

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