Factorising Quadratics with a Coefficient of x Squared Greater than 1

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:

  • 12x² + x – 6
  • 4x² – 4x – 3
  • 6x² + 5x – 3

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:

  • Add up to the coefficient of the x term (5 in this case).
  • Multiply to get the product of the coefficient of x² (2) and the constant term (-3), which is -6.

After trying out a few possibilities, we find that the numbers 6 and -1 fit these criteria:

  • Sum: 6 + (-1) = 5
  • Product: 6 × (-1) = -6

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.

Examples

Example 1:

Factorise 12x² + x – 6

We need two numbers with a sum of 1 and a product of 12(-6) = -72. These numbers are 9 and -8. So, we rewrite the expression as:

12x² + 9x – 8x – 6

Factorising, we get:

12x² + 9x – 8x – 6 = 3x(4x + 3) – 2(4x + 3)

= (3x – 2)(4x + 3)

Example 2:

Factorise 4x² – 4x – 3

We need two numbers with a sum of -4 and a product of 4(-3) = -12. These numbers are -6 and 2. So, we rewrite the expression as:

4x² – 6x + 2x – 3

Factorising, we get:

4x² – 6x + 2x – 3 = 2x(2x – 3) + 1(2x – 3)

= (2x + 1)(2x – 3)

Example 3:

Factorise 20x² + 3x – 9

We need two numbers with a sum of 3 and a product of 20(-9) = -180. These numbers are 15 and -12. So, we rewrite the expression as:

20x² – 12x + 15x – 9

Factorising, we get:

20x² – 12x + 15x – 9 = 4x(5x – 3) + 3(5x – 3)

= (4x + 3)(5x – 3)

Example 4:

Factorise:

i) 3x² – 27

ii) 5x² 20

i) To factorise this expression, first find the common factor, which is 3:

3x² – 27 = 3(x² – 9)

Now, we can see that the expression inside the parenthesis is a difference of squares:

= 3(x + 3)(x – 3)

ii) Similarly, first find the common factor, which is 5:

5x² – 20 = 5(x² – 4)

Then, factorise the difference of squares:

= 5(x + 2)(x – 2)

Example 5:

Factorise:

i) x² – 121

ii) x²y – 121y

iii) 6x²y – 726y

i) This expression is a difference of squares:

x² – 121 = (x + 11)(x – 11)

ii) First, find the common factor, which is y:

x²y – 121y = y(x² – 121)

Then, factorise the difference of squares:

= y(x + 11)(x – 11)

iii) First, find the common factor, which is 6y:

6x²y – 726y = 6y(x² – 121)

Finally, factorise the difference of squares:

= 6y(x + 11)(x – 11)$

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