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Quadratics in algebraic fractions

An example of an algebraic fraction with a quadratic expression in the numerator. In this case, it is x squared plus 2x minus 3.

Let's take a look at algebraic fractions where the denominator is a quadratic expression. Such fractions are common in mathematics and engineering.

Before you continue with the content on this page, it is important that you are confident about identifying common factors, factorising quadratics using common factors and the difference of two squares rule.

The best way to grasp how to work with quadratics in algebraic fractions is through examples.

Example – adding or subtracting algebraic fractions with quadratics

Simplify 2x2+x121x29.

You can multiply the denominators as is to form the lowest common denominator, but this would give you an x4. But since this can be tricky to work with, you can factorise the denominators instead.

x2+x12=(x+4)(x3)x29=(x+3)(x3)

We can rewrite the original operation as:

2x2+x121x29=2(x+4)(x3)1(x3)(x+3)

A common denominator is (x+4)(x3)(x+3).

To write the individual fractions with this common denominator, we need to multiply 2 by (x+3), and 1 by (x+4).

=2(x+3)(x+4)(x3)(x+3)1(x+4)(x+4)(x3)(x+3)=2(x+3)1(x+4)(x+4)(x3)(x+3)=2x+6x4(x+4)(x3)(x+3)=x+2(x+4)(x3)(x+3)where x4,3,3

Exercise – adding or subtracting algebraic fractions with quadratics

Simplify 2x2+2x+1x24.

3x4x(x+2)(x2)where x2,0,2

Example – dividing algebraic fractions with quadratics

Simplify 12x1+2x.

=(12x)÷(1+2x)convert each into single fractions=x2x÷x+2xchange sign and invert last fraction=x2x×xx+2divide top and bottom by 1=x2x+2where x2,0

Exercise – dividing algebraic fractions with quadratics

Simplify 16xx23.

2xwhere x0,6

Example – simplifying algrebraic fractions with quadratics and mixed operations

Simplify 3x2+2x3+1x13x1+3.

This expression involves nested fractions. We can step through it by looking at the numerator and denominator separately.

Let's look at the numerator first. You can multiply the denominators as is to form the lowest common denominator, but this would give you an x4. This can be tricky to work with. Instead, you can factorise the denominators.

x2+2x3=(x+3)(x1)

We can rewrite the original numerator as:

3x2+2x3+1x1=3(x+3)(x1)+1x1=3(x+3)(x1)+1x1

A common denominator is (x+3)(x1).

To write the individual fractions with this common denominator, we need to multiply 1 by (x+3).

=3(x+3)(x1)+(x+3)(x+3)(x1)=3+x+3(x+3)(x1)=x(x+3)(x1)

Now, we can look at the denominator. To make a common denominator, we need to multiply 3 by (x1).

3x1+3=3x1+3(x1)x1=3xx1

Putting the numerator and denominator together, we get:

x(x+3)(x1)3xx1

We can rewrite the fraction using a division symbol and complete the simplification.

=x(x+3)(x1)÷3xx1change sign and invert last fraction=x(x+3)(x1)×x13divide top and bottom by (x1)=x3x(x+3)divide top and bottom by x=13(x+3)where x3,0,1

Exercise – simplifying algebraic fractions with quadratics and mixed operations

Simplify 1x24+12x+41+2x2.

12(x+2)where x2,0,2

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Further resources

Factorisation

Use this resource if you need a refresher on how to identify common factors.

Quadratic factorisation

Use this resource if you need a refresher on how to factorise quadratic expressions using common factors.

Difference of two squares

Use this resource if you need a refresher on the difference of two squares (DOTS) rule.