To factorise quadratic expressions of the form ax² + bx + c, we look for two numbers with a sum equal to the coefficient of the linear term (b) and a product equal to the product of the quadratic coefficient (a) and the constant term (c).
After this, we rewrite the middle term (bx) in the expression ax² + bx + c as the sum of these two numbers and factorise the resulting expression.
However, sometimes, we need to factorise quadratic expressions where the coefficient of x² is not one. Some examples include:
Let’s look at how to factorise these expressions:
Consider the quadratic expression 2x² + 5x – 3. Here, the coefficient of x² is 2.
To factorise this expression, we first look for two numbers that:
After trying out a few possibilities, we find that the numbers 6 and -1 fit these criteria:
Now, we rewrite the middle term (5x) as the sum of these two numbers (6x and -1x):
2x² + 5x – 3 = 2x² + 6x – 1x – 3
Next, we factorise the expression by grouping:
2x² + 6x – 1x – 3 = 2x(x + 3) – 1(x + 3)
Finally, we factor out the common factor (x + 3):
2x(x + 3) – 1(x + 3) = (2x – 1)(x + 3)
So, the factorised form of 2x² + 5x – 3 is (2x – 1)(x + 3).
Let’s look at some more examples.
Factorise 12x² + x – 6
Factorise 4x² – 4x – 3
Factorise 20x² + 3x – 9
i) 3x² – 27
ii) 5x² – 20
i) x² – 121
ii) x²y – 121y
iii) 6x²y – 726y