Expanding brackets calculator with working

Each term in the first bracket multiplies each term in the second. FOIL (First, Outside, Inside, Last) is one way to remember the four products:

For $(x + 3)(x + 5)$:

• First: $x \times x = x^2$
• Outside: $x \times 5 = 5x$
• Inside: $3 \times x = 3x$
• Last: $3 \times 5 = 15$

Collect: $x^2 + 8x + 15$. A grid with one bracket along the top and one down the side works the same way.

$(x + 3)^2$ means $(x + 3)(x + 3)$, which is $x^2 + 6x + 9$, not $x^2 + 9$. The middle term comes from the two "outside" and "inside" products. In general, $(x + a)^2 = x^2 + 2ax + a^2$.

Expand two brackets first, then multiply the result by the third.

$(x + 1)(x + 2)(x + 3)$: first $(x + 1)(x + 2) = x^2 + 3x + 2$. Then $(x^2 + 3x + 2)(x + 3) = x^3 + 3x^2 + 3x^2 + 9x + 2x + 6 = x^3 + 6x^2 + 11x + 6$.

Multiply the term outside by everything inside: $3(x + 4) = 3x + 12$, and $2x(x - 5) = 2x^2 - 10x$. In the calculator, enter a single term like 3 as a bracket with $a = 0$ and $b = 3$.

• Missing the middle terms when squaring a bracket.
• Sign errors: a negative times a negative is positive.
• Not collecting like terms at the end.

Practise with our expanding brackets questions, and reverse the process with the factorising calculator.

• Write all four products before collecting; marks are given for them.
• "Expand and simplify" means collect like terms too.
• Triple brackets are Higher tier at GCSE.

Expanding brackets is the first step in many longer questions:

• Solving equations: $3(x - 2) = 2(x + 1)$ becomes $3x - 6 = 2x + 2$, so $x = 8$.
• Proving identities: show that $(n + 1)^2 - n^2 = 2n + 1$ by expanding: $n^2 + 2n + 1 - n^2 = 2n + 1$.
• Area problems: a rectangle with sides $(x + 3)$ and $(x + 5)$ has area $x^2 + 8x + 15$.

Our algebraic proof questions use this a lot.

Negative signs cause most errors, so take them one product at a time.

Example: $(x - 4)(x - 3)$. The four products are $x^2$, $-3x$, $-4x$ and $(-4) \times (-3) = +12$. Collected: $x^2 - 7x + 12$.

Example: $(3 - x)(x + 2)$. Products: $3x$, $6$, $-x^2$, $-2x$. Collected and written with the highest power first: $-x^2 + x + 6$.

A minus in front of a bracket changes every sign inside: $-(2x - 5) = -2x + 5$.