Factorising the expression `9x^2 - 25x - 6` includes a coefficient for the `x^2` term (9).
This makes determining the factors for the quadratic more complicated.
The method is to
1. multiply the `x^2` coefficient and the integer;
2. determine the factors from the first step that add up to the `x` term coefficient;
3. rewrite the `x` term using these factors;
4. factorise the expression to get two brackets that are the same;
5. rewrite the expression with this new factorisation.
Factorise `8x^2 + 22x + 5`
(step 1) Multiply 8 and 5 to get 40
(step 2) Determine the factors of 40 that add to 22: 2 and 20
(step 3) Rewrite the expression as `8x^2 + 2x + 20x + 5`
(step 4) Factorise the expression `2x(4x + 1) + 5(4x + 1)`
(step 5) Write the expression in factorised form `(2x + 5)(4x + 1)`
(check) Multiply out the factorised expression to get `8x^2 + 2x + 20x + 5`
Answer: `(2x + 5)(4x + 1)`
Factorise `-21x^2 + 29x - 10`
(step 1) Multiply -21 and -10 to get 210
(step 2) Determine the factors of 210 that also add to 29: 14 and 15
(step 3) Rewrite the expression as `-21x^2 + 14x + 15x - 10`
(step 4) Factorise the expression `7x(-3x + 2) + 5(3x - 2)`
Adjust to make the terms in the brackets the same `7x(-3x + 2) - 5(-3x + 2)`
(step 5)Write the expression in factorised form `(7x - 5)(-3x + 2)`
(check) Multiply out to get `-21x^2 +15x + 14x -10`
Answer: `(-7x + 5)(3x - 2)`
See also Factorising Quadratic Expressions