Question

Factorise the following:

`x^2 + 7/6 x - 5/6`

Answer 1

Given expression: `x^2 + 7/6 x - 5/6`

Step-wise calculation:

Step 1: Clear the fractions by multiplying through by 6 to make factorisation easier:

`6 xx (x^2 + 7/6x - 5/6) = 6x^2 + 7x - 5`

Step 2: Factorise the quadratic 6x² + 7x – 5.

We look for two numbers whose product is 6 × (–5) = –30 and whose sum is 7. 

Those numbers are 10 and –3 since 10 × –3 = –30 and 10 + (–3) = 7.

Step 3: Rewrite the middle term 7x as 10x – 3x: 

6x² + 10x – 3x – 5

Step 4: Group terms:

(6x² + 10x) – (3x + 5)

Step 5: Factor each group:

2x(3x + 5) – 1(3x + 5)

Step 6: Factor out the common binomial factor:

(3x + 5)(2x – 1)

Step 7: Since we multiplied the original expression by 6, to factorise the original expression, put the factor `1/6` back:

\[ x^2 + \frac{7}{6}x - \frac{5}{6} = \frac{1}{6}(3x + 5)(2x - 1) \]