Expanding double brackets is a core algebra skill that helps you multiply expressions like $(x + 1)(x + 2)$. It’s essential for understanding quadratic equations and is used in everything from calculating the area of a garden to designing computer graphics.
The Two Main Methods
The goal is to make sure every term in the first bracket is multiplied by every term in the second. Both methods will get you the right answer — pick the one that makes the most sense to you!
Method 1: The FOIL Method
FOIL is an acronym to help you remember the four multiplications you need to do:
- First: Multiply the first terms.
- Outer: Multiply the outer terms.
- Inner: Multiply the inner terms.
- Last: Multiply the last terms.
Then, you add the results and collect any like terms to simplify.
Method 2: The Grid Method
This is a visual way to ensure you don’t miss any terms. Write the terms of one bracket along the top and the other down the side, then multiply to fill each cell.
Multiply to fill in each cell, then add the terms and simplify.
Worked Examples
Example 1: All Positives
Expand and simplify $(x + 3)(x + 5)$.
FOIL Method:
- First: $x \times x = x^2$
- Outer: $x \times 5 = 5x$
- Inner: $3 \times x = 3x$
- Last: $3 \times 5 = 15$
Combine: $x^2 + 5x + 3x + 15$
Answer: $x^2 + 8x + 15$
Example 2: With Negatives
Expand and simplify $(y + 4)(y – 6)$.
FOIL Method:
- First: $y \times y = y^2$
- Outer: $y \times (-6) = -6y$
- Inner: $4 \times y = 4y$
- Last: $4 \times (-6) = -24$
Combine: $y^2 – 6y + 4y – 24$
Answer: $y^2 – 2y – 24$
Example 3: Squaring a Bracket
Expand and simplify $(x + 6)^2$.
First, rewrite as $(x + 6)(x + 6)$.
FOIL Method:
- First: $x \times x = x^2$
- Outer: $x \times 6 = 6x$
- Inner: $6 \times x = 6x$
- Last: $6 \times 6 = 36$
Combine: $x^2 + 6x + 6x + 36$
Answer: $x^2 + 12x + 36$
Example 4: With Coefficients
Expand and simplify $(3y – 1)(2y + 5)$.
FOIL Method:
- First: $3y \times 2y = 6y^2$
- Outer: $3y \times 5 = 15y$
- Inner: $(-1) \times 2y = -2y$
- Last: $(-1) \times 5 = -5$
Combine: $6y^2 + 15y – 2y – 5$
Answer: $6y^2 + 13y – 5$
Tutor Insights
🤔 Common Misunderstandings
- Squaring a bracket: Thinking $(x+3)^2$ is $x^2 + 9$. This is wrong! You must write it as $(x+3)(x+3)$ and expand it fully to get the middle term.
- Dealing with negatives: Forgetting that a negative times a negative is a positive, or making sign errors when collecting the middle terms.
📝 Common Exam Mistakes
- Forgetting to simplify: Getting all four terms correct but then forgetting to combine the two middle ‘x’ terms.
- Missing terms: Accidentally skipping one of the FOIL steps, usually the Outer or Inner multiplication. The Grid Method helps prevent this.
- Sign errors: The number one cause of lost marks! Be extra careful when multiplying or adding negative terms.
Practice Questions
Expand and simplify the following expressions.
- $(x + 2)(x + 7)$
- $(y – 3)(y + 5)$
- $(a – 4)(a – 6)$
- $(2x + 1)(x + 5)$
- $(x + 9)^2$
Show Answers
- Answer: $x^2 + 9x + 14$
- Answer: $y^2 + 2y – 15$
- Answer: $a^2 – 10a + 24$
- Answer: $2x^2 + 11x + 5$
- Answer: $x^2 + 18x + 81$
FAQs
Q: Why isn’t $(x+2)^2$ equal to $x^2 + 4$?
A: Because squaring means multiplying by itself, so $(x+2)^2 = (x+2)(x+2)$. When you expand this, you get $x^2 + 2x + 2x + 4$, which simplifies to $x^2 + 4x + 4$. You get a middle term that’s missed if you just square each part individually.
Q: Is the Grid Method or FOIL better?
A: Neither is “better” — they both achieve the same result! The Grid Method is often more reliable for preventing mistakes, especially with negatives. FOIL can be quicker once you’re confident. Try both and see which you prefer.
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