Further Quadratic Expressions

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.

Example 1

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: $(-7x + 5)(3x - 2)$