How would I simplify (3x^2 − 8x − 3)/(2x^2 −6x) fully?
The above sample question was taken from the 2018 GCSE Maths Higher paper.First we simplify the numerator, 3x^2 - 8x - 3, which is a quadratic written in the form ax^2 + bx + c so we can simplify it by factorising.If we multiply the a and c terms of the quadratic (3*-3) we get -9, so need to find two numbers that multiply to make -9 and add up to make the b term of the quadratic, here -8. Here this would be -9 and 1 because -9*1 is -9 and -9+1 is -8.We can now think of this same quadratic then as 3x^2 - 9x + 1x - 3. The first part 3x^2 - 9x can be simplified by taking out a factor of 3x making the numerator 3x(x-3) + 1(x - 3) so the factorised numerator would be (3x+1)(x-3)Next we simplify the denominator: 2x^2 - 6xThe common factor that can be "taken out" is 2x making the denominator 2x(x - 3)The factorised fraction is then (3x + 1)(x - 3)/2x(x - 3)As you can see (x - 3) is on the top and bottom so as this is a fraction we can cancel this out making the simplified fraction (3x + 1)/2x. This fraction is now simplified fully as top and bottom can no longer be divided by any common factor.
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