Multiplication of two linear expressions

Normally when dealing with brackets, we use and calculate the value of the bracket before performing any other operations.

Example

(3 + 4) × (8 – 5) = (7) × (3) = 21

However, when we have algebraic terms in brackets, this is no longer possible. Consider the following expression which we must simplify:

(\({a}\) + \({b}\))(\({c}\) + \({d}\))

As we cannot simplify what is within the bracket, we must instead use the distributive property of this expression. This means that we can rewrite the above expression in the following way:

(\({a}\) + \({b}\))(\({c}\) + \({d}\)) = \({ac}\) + \({ad}\) + \({bc}\) + \({bd}\)

Look closely, as we have removed the brackets in the expression by multiplying both terms in the first bracket by both terms of the second bracket. This process is called expansion.

Using the same numerical example as before, we would have:

(3 + 4) × (8 – 5)

= (3 × 8) + (3 × -5) + (4 × 8) + (4 × -5)

= (24) + (-15) + (32) + (-20)

= 21

We can see that while this method of calculation works for numbers, it is far longer and more complicated than normal.

Example one

Expand the following expression:

(2\({x}\) + 3)(\({x}\) - 2)

Using the distributive property we have:

(2\({x}\) + 3)(\({x}\) - 2)

= (2\({x}\) × \({x}\)) + (2\({x}\) × -2) + (3 × \({x}\)) + (3 × -2)

= 2\({x^2}\) -4\({x}\) + 3\({x}\) -6

= 2\({x^2}\) - \({x}\) - 6

We have successfully expanded the brackets.

Example two

Expand the following expression:

(-3\({x}\) + 4)²

First we must realise that:

(-3\({x}\) + 4)² = (-3\({x}\) + 4) × (-3\({x}\) + 4)

Then, as before:

(-3\({x}\) + 4) × (-3\({x}\) + 4) = (-3\({x}\) × -3\({x}\)) + (4 × -3\({x}\)) + (-3 × 4) + (4 × 4)

= 9\({x^2}\) - 12\({x}\) - 12\({x}\) + 16

= 9\({x^2}\) - 24\({x}\) + 16

Example three

Expand the following expression:

(\({x}\) + 7)(\({x}\) - 7)

As before:

(\({x}\) + 7) (\({x}\) - 7) = \({x^2}\) + 7\({x}\) - 7\({x}\) - 49

= \({x^2}\) - 49

You will notice here that the answer is the difference between the square of the two numbers in the brackets: \({x}\) and 7. This is in fact a special type of algebraic expansion called the difference between two squares. The difference of two squares is the answer when we expand an expression in the form (\({a}\) + \({b}\))(\({a}\) – \({b}\)).

Question

Expand the following expression:

(\({x}\) + 3)(\({x}\) - 3)

Question

Expand the following expression:

(4\({x}\) + 3)²

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Quadratic expressions - Intermediate & Higher tier – WJECMultiplication of two linear expressions