Question

Factorise the following:

`3sqrt(2)x^2 - 11x + 4sqrt(2)`

Answer 1

We need to factorise:

`3sqrt(2)x^2 - 11x + 4sqrt(2)`

Step 1: Multiply the coefficient of x² and the constant term

`3sqrt(2) xx 4sqrt(2) = 12 xx 2`

`3sqrt(2) xx 4sqrt(2) = 24`

So we need two numbers whose product = 24 and sum = –11.

Those numbers are –3 and –8.   ...(Because –3 × –8 = 24 – 3 and –3 + –8 = –11)

Step 2: Split the middle term

`3sqrt(2)x^2 - 3x - 8x + 4sqrt(2)`

Step 3: Factor by grouping

Group the terms:

`(3sqrt(2)x^2 - 3x) - (8x - 4sqrt(2))`

Factor each group:

`3x(sqrt(2)x - 1) - 4(2x - sqrt(2))`

Write the second bracket in the same form as the first:

`2x - sqrt(2) = sqrt(2)(sqrt(2)x - 1)`

So, `-4(2x - sqrt(2)) = -4sqrt(2)(sqrt(2)x - 1)`

Thus the expression becomes `3x(sqrt(2)x - 1) - 4sqrt(2)(sqrt(2)x - 1)`

Step 4: Factor out the common bracket

`(sqrt(2)x - 1)(3x - 4sqrt(2))`