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Year 10 Interactive Maths - Second Edition


Factors of Quadratic Trinomials of the Type ax + bx + c

Factorise (factorize) the quadratic trinomial by experimenting with its linear factors using the cross multiplication method.

The linear factors (7x + 1) and (x + 2) do not produce the middle term of 9x.

 7x2 + x1 = 14x+ x = 15x.  We reject this pair as the middle term is 9x.

The linear factors (7x + 2) and (x + 1) produce the middle term of 9x.

So 7x^2 + 9x + 2 = (7x+2)(x+1)  

Note:

The solution is read across.


Use of a Common Factor

If a quadratic trinomial has a common factor, take it out and place it in front of the brackets.  Then use the cross-multiplication method to factorise the quadratic trinomial.

Example 10

Factorise (factorize) 12x^2 - 14x - 6.

Solution:
Take out the common factor of 2 and then factorise (factorize) the quadratic trinomial by experimenting with its linear factors using the cross multiplication method.

The factors 6x and x don't produce the middle term.

The linear factors (2x - 3) and (3x + 1) produce the middle term -7x.

So 12x^2 - 14x - 6 = 2(2x-3)(3x+1)

Check:
Check by expanding 2(2x-3)(3x+1).
Note:

The common factor 2 was taken out first to make the factorisation simpler.


Example 11

Factorise (factorize) 18a^2 + 3a - 6.

Solution:

Take out the common factor of 3 and then factorise (factorize) the quadratic trinomial by experimenting with its linear factors using the cross multiplication method.

The factors 6a and a don't produce the middle term.

Now try the factors 2a and 3a.

The linear factors (2a - 1) and (3a + 2) produce the middle term a.

So 18a^2 + 3a - 6 = 3(2a-1)(3a+2)

Check:

Check by expanding 3(2a-1)(3a+2).


Note:

The common factor 3 was taken out first to make the factorisation simpler.


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