Question

Factorise the following:

`2sqrt(3)x^2 + x - 5sqrt(3)`

Answer 1

Given: `2sqrt(3)x^2 + x - 5sqrt(3)`

Step-wise calculation:

Step 1: Find two numbers to split the middle term

The given expression is `2sqrt(3)x^2 + x - 5sqrt(3)`.

The coefficients are `a = 2sqrt(3), b = 1` and `c = - 5sqrt(3)`.

We calculate the product ac: 

`ac = (2sqrt(3))(-5sqrt(3))`

ac = –10 × 3

ac = –30

We need to find two numbers that multiply to –30 and sum to b = 1.

These numbers are 6 and –5.

Step 2: Split the middle term and group the terms

We split the middle term, x, into 6x – 5x:

`2sqrt(3)x^2 + 6x - 5x - 5sqrt(3)`

Now we group the terms:

`(2sqrt(3)x^2 + 6x) + (-5x - 5sqrt(3))`

Step 3: Factor the Greatest Common Factor (GCF) from each group

We factor the GCF from the first group.

Note that 6 = 2 × 3

`6 = 2 xx (sqrt(3))^2`

 The GCF is `2sqrt(3)x`:

`2sqrt(3)x (x + sqrt(3)) + (-5x - 5sqrt(3))`

Factor –5 from the second group:

`2sqrt(3)x (x + sqrt(3)) - 5(x + sqrt(3))`

Step 4: Factor out the common binomial

Factor out the common binomial factor `(x + sqrt(3))` from the entire expression:

`(x + sqrt(3))(2sqrt(3)x - 5)`