Year 9

Expand Double Brackets

Expand the product of two linear expressions such as $(x+a)(x+b)$ and simplify by collecting like terms to give a quadratic in the form $ax^2 + bx + c$ .

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Lesson overview

Algebra - expanding and factorising

When multiplying two brackets, every term in the first bracket multiplies every term in the second. Use FOIL: First · First, Outer · Outer, Inner · Inner, Last · Last. $(x+a)(x+b) = x^2 + bx + ax + ab = x^2 + (a+b)x + ab$ After expanding, collect like terms to write the result in the form $ax^2 + bx + c$ .

FOIL: multiply every pair

Expand (x + 3)(x + 5).

First $x × x = {x}^{2}$

Outerx × 5 = 5x

Inner3 × x = 3x

Last3 × 5 = 15

Collect the two middle terms: 5x + 3x = 8x. The final answer is always a trinomial (three terms).

Negative terms — take care with signs

Expand (x + 4)(x − 2).

First $x × x = {x}^{2}$

Outerx × (−2) = −2x

Inner4 × x = 4x

Last4 × (−2) = −8

Negative × positive = negative. Collect: −2x + 4x = 2x.

Both terms negative

Expand (x − 3)(x − 7).

Negative × negative = positive. The constant (last term) is always positive when both signs are negative.

With coefficients — multiply carefully

Expand (2x + 1)(3x − 4).

First $2x × 3x = 6{x}^{2}$

Outer2x × (−4) = −8x

Inner1 × 3x = 3x

Last1 × (−4) = −4

When there are coefficients, the ${x}^{2}$ term is no longer just ${x}^{2}$ . Multiply both numbers and both x-terms.

Worked example

A rectangle has width (x + 2) cm and length (x + 6) cm. Find an expression for its area. Expand and simplify.

F $x × x = {x}^{2}$

Ox × 6 = 6x

I2 × x = 2x

L2 × 6 = 12

Area of a rectangle = width × length.
Use FOIL to expand (x + 2)(x + 6).
First: x × x = ${x}^{2}$ .
Outer: x × 6 = 6x.
Inner: 2 × x = 2x.
Last: 2 × 6 = 12.
Collect like terms: 6x + 2x = 8x.
Area = ${x}^{2}$ + 8x + 12 ${cm}^{2}$ .

Try it

Use the examples carefully, then choose the answer.

Question 1

Expand and simplify (x + 2)(x + 7).

💡 Use FOIL. Collect the two middle terms.

Question 2

Expand and simplify (x + 5)(x − 3).

💡 Be careful with signs. 5 × (−3) = −15.

Question 3

Expand and simplify (x − 4)(x − 6).

💡 Negative × negative = positive for the last term.

Question 4

Expand and simplify (x + 8)(x − 8).

💡 The two middle terms cancel. This is called the difference of two squares.

Question 5

Expand and simplify (2x + 3)(x + 4).

💡 First = 2x × x = 2 ${x}^{2}$ . Collect: 8x + 3x.

Expand Double Brackets

Lesson overview

Worked example

Try it

Independent practice will plug in here