Algebra
Grade 5-6Expanding Double Brackets
Expanding double brackets means multiplying every term in the first bracket by every term in the second, then collecting the like terms that result. This lesson covers expanding two brackets, expanding brackets that contain negative terms, expanding a squared bracket such as (x + 5)², and expanding brackets where one term has a coefficient in front of x.
What you need to know
- Multiply each term in the first bracket by each term in the second (FOIL, or a grid).
- (x + 3)(x + 5) gives x squared plus 8x plus 15.
- The x term comes from adding the two products; the number term from multiplying them.
- (x + a) squared means (x + a)(x + a), which is x squared plus 2ax plus a squared, not x squared plus a squared.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Expanding two brackets
Multiply each term in the first bracket by each term in the second bracket, then collect the like terms.
(x + 3)(x + 5) expands to x × x + x × 5 + 3 × x + 3 × 5, which is x² + 5x + 3x + 15. Collecting the like terms gives x² + 8x + 15.
Expanding brackets with negative terms
Multiply through in exactly the same way, keeping careful track of each sign, remembering that multiplying two negative terms gives a positive result.
(x − 3)(x − 6) expands to x² − 6x − 3x + 18. Since −3 × −6 = 18, collecting the like terms gives x² − 9x + 18.
Expanding a squared bracket
A bracket squared means the bracket multiplied by itself, so write it out as two identical brackets before expanding, rather than just squaring each term.
(x + 5)² means (x + 5)(x + 5), which expands to x² + 5x + 5x + 25. Collecting the like terms gives x² + 10x + 25.
Expanding brackets with a coefficient
When one of the terms has a coefficient in front of x, multiply through in the same way, taking extra care when collecting the two middle terms, since they may have different coefficients.
(2x + 3)(x + 4) expands to 2x² + 8x + 3x + 12. Collecting the middle terms, 8x + 3x = 11x, gives 2x² + 11x + 12.
Worked examples
Three exam-style questions, fully solved
Expand and simplify (x + 3)(x + 5).
Easy- 1.Multiply each term in the first bracket by each term in the second: x² + 5x + 3x + 15
- 2.Collect the like terms: 5x + 3x = 8x
Answer: x² + 8x + 15
Expand and simplify (x + 4)(x − 2).
Medium- 1.Multiply each term in the first bracket by each term in the second: x² − 2x + 4x − 8
- 2.Collect the like terms: −2x + 4x = 2x
Answer: x² + 2x − 8
Expand and simplify (x + 5)².
Hard- 1.Write the bracket squared as two identical brackets: (x + 5)(x + 5)
- 2.Multiply each term in the first bracket by each term in the second: x² + 5x + 5x + 25
- 3.Collect the like terms: 5x + 5x = 10x
Answer: x² + 10x + 25
Practice
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Expand and simplify (x + 3)(x + 4).
Expand and simplify (x + 5)(x − 2).
Expand and simplify (x − 6)(x − 3).
Expand and simplify (x + 4)².
Expand and simplify (2x + 1)(x + 5).
Expand and simplify (3x − 2)(2x + 3).
Avoid these
The mistakes students make most often
Forgetting one of the four multiplications when expanding, most often the "outer" or "inner" pair of terms.
Getting a sign wrong when one or both brackets contain a negative term, especially forgetting that two negative terms multiply to give a positive result.
Expanding a squared bracket like (x + 5)² as x² + 25, instead of writing it as (x + 5)(x + 5) and expanding fully to get x² + 10x + 25.
Forgetting to collect the two middle terms into a single term after expanding, and leaving the answer as four separate terms.
When a coefficient appears in front of x in one of the brackets, making an error combining the two middle terms, since they no longer have matching coefficients.
FAQ
Questions students and parents ask
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