Mathematics
Special Products: Formulas and 8 Solved Exercises
The square of a binomial is (a+b)² = a² + 2ab + b², never a² + b². Get every formula, a mistakes table and 8 solved exercises, step by step with checks.
Recommended for: Grade 9 · Grade 10
Special products are products of polynomials that follow a fixed pattern, so you can write the result at once without multiplying term by term. The best known is the square of a binomial: (a + b)² = a² + 2ab + b². You will use these patterns throughout algebra, in factoring, equations and algebraic fractions.
The formulas you need
| Name | Formula | Terms in result |
|---|---|---|
| Square of a sum | (a + b)² = a² + 2ab + b² | 3 |
| Square of a difference | (a − b)² = a² − 2ab + b² | 3 |
| Difference of squares | (a + b)(a − b) = a² − b² | 2 |
| Cube of a sum | (a + b)³ = a³ + 3a²b + 3ab² + b³ | 4 |
| Cube of a difference | (a − b)³ = a³ − 3a²b + 3ab² − b³ | 4 |
| Square of a trinomial | (a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc | 6 |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) | product |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) | product |
Square of a binomial
The square of a binomial is the square of the first term, plus twice the product of the two terms, plus the square of the second term. The middle term takes the sign of the binomial: plus for a sum, minus for a difference. The two squares are always positive.
Example: (2x + 7)² = 4x² + 28x + 49. The middle term is 2 · 2x · 7 = 28x.
Why is there a middle term? A square with side a + b splits into a square of area a², a square of area b², and two rectangles of area a · b. That is where 2ab comes from. If you forget it, you lose two whole rectangles.
Difference of squares
When the two factors have the same terms and opposite signs, the middle terms cancel and you are left with the difference of the squares: (A + B)(A − B) = A² − B². Example: (5y + 2)(5y − 2) = 25y² − 4.
Watch the order. The result is always “square of the positive term minus square of the negative term.” For (3 − x)(3 + x) the answer is 9 − x², not x² − 9.
Cube of a binomial
The coefficients 1, 3, 3, 1 come from the fourth row of Pascal’s triangle. The powers of a go down from 3 to 0, and the powers of b go up from 0 to 3. For a difference the signs alternate: +, −, +, −.
Example: (2x + 1)³ = 8x³ + 3 · 4x² · 1 + 3 · 2x · 1 + 1 = 8x³ + 12x² + 6x + 1.
Square of a trinomial
Square all three terms, then add the three doubled products, each with the sign of its two terms. Example: (a + 2b − 3)² = a² + 4b² + 9 + 4ab − 6a − 12b.
To check, plug in a = 1 and b = 1. On the left (1 + 2 − 3)² = 0, and on the right 1 + 4 + 9 + 4 − 6 − 12 = 0.
Using them backwards
Read from right to left, the same formulas let you factor. Look for a trinomial with two perfect squares and a doubled product: it is the square of a binomial. Look for a difference of two squares: it factors as sum times difference. If powers still feel shaky, review the laws of exponents. To use these expansions inside equations, practice linear equations and then quadratic equations.
The most common mistakes
| Mistake | Why it is wrong | Fix |
|---|---|---|
| (a + b)² = a² + b² | The middle term is missing | a² + 2ab + b² |
| (a − b)² = a² − b² | Mixed up with the difference of squares | a² − 2ab + b² |
| (3x)² = 3x² | The coefficient is not squared | 9x² |
| (a − b)² = a² − 2ab − b² | The square of b is always positive | a² − 2ab + b² |
| (a + b)³ = a³ + b³ | The two mixed terms are missing | a³ + 3a²b + 3ab² + b³ |
| − (x − 2)² expanded as − x² − 4x + 4 | The minus in front flips every sign | − x² + 4x − 4 |
Edge cases to watch
- Coefficients and powers of letters: in (2x²y)² the square is 4x⁴y², because the exponents double.
- Negative first term: in (−x + 3)² the first term is −x and its square is x². The expansion is x² − 6x + 9.
- Fractions: (x/2 + 1)² = x²/4 + x + 1, because the middle term is 2 · x/2 · 1 = x.
- Several squares in one expression: expand each special product separately, then combine like terms.
A four-step method
- Recognize the form: how many terms should the result have, and what signs do the factors have?
- Identify A and B, putting them in parentheses when they have coefficients.
- Apply the formula without skipping any term.
- Check with a simple value, such as x = 1 or x = 2.
In the exercises below the difficulty grows, from the basic formulas to mental math, factoring and equations. Every result is checked.
Solved exercises
1. Expand (x + 5)². base
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- This is the square of a binomial: (a + b)² = a² + 2ab + b², with a = x and b = 5.
- Square of the first term: x².
- Twice the product: 2 · x · 5 = 10x.
- Square of the second term: 5² = 25.
Answer: x² + 10x + 25
2. Expand (3a − 2)². base
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- Use (A − B)² = A² − 2AB + B², with A = 3a and B = 2.
- A² = (3a)² = 9a². The coefficient 3 is squared along with the letter.
- Twice the product: 2 · 3a · 2 = 12a, with a minus sign.
- B² = 4.
Answer: 9a² − 12a + 4
3. Expand (4x + 3)(4x − 3). base
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- The factors have the same terms and opposite signs, so this is a difference of squares: (A + B)(A − B) = A² − B².
- A = 4x, so A² = 16x².
- B = 3, so B² = 9.
- The middle terms cancel, so no term in x remains.
Answer: 16x² − 9
4. Sam writes (x + 3)² = x² + 9. Find the mistake, write the correct expansion, and find how far apart the two results are when x = 2. intermedio
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- Sam squared each term and forgot the middle term 2 · x · 3 = 6x.
- Correct expansion: (x + 3)² = x² + 6x + 9.
- For x = 2 the correct value is 4 + 12 + 9 = 25, which matches (2 + 3)² = 25.
- Sam's result gives 4 + 9 = 13.
- The difference is 25 − 13 = 12, exactly the missing term 6x at x = 2.
Answer: The middle term is missing: (x + 3)² = x² + 6x + 9. At x = 2 you get 25 versus 13, a difference of 12.
5. Compute 99² and 51 · 49 in your head, without a calculator. intermedio
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- Write 99 = 100 − 1, so 99² = (100 − 1)² = 100² − 2 · 100 · 1 + 1² = 10,000 − 200 + 1 = 9,801.
- Write 51 = 50 + 1 and 49 = 50 − 1. The product is a difference of squares.
- 51 · 49 = (50 + 1)(50 − 1) = 50² − 1² = 2,500 − 1 = 2,499.
Answer: 99² = 9,801; 51 · 49 = 2,499
6. Expand (x − 2)³. intermedio
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- Use (a − b)³ = a³ − 3a²b + 3ab² − b³, with a = x and b = 2.
- a³ = x³.
- 3a²b = 3 · x² · 2 = 6x², with a minus sign.
- 3ab² = 3 · x · 4 = 12x, with a plus sign.
- b³ = 8, with a minus sign.
- Check with x = 3: (3 − 2)³ = 1 and 27 − 54 + 36 − 8 = 1.
Answer: x³ − 6x² + 12x − 8
7. Factor 25x² − 30x + 9 and 49a² − 16b⁴. avanzato
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- First polynomial: 25x² = (5x)² and 9 = 3² are perfect squares.
- The middle term should be 2 · 5x · 3 = 30x. It matches, with a minus sign.
- So 25x² − 30x + 9 = (5x − 3)².
- Second polynomial: 49a² = (7a)² and 16b⁴ = (4b²)², since (b²)² = b⁴. It is a difference of squares.
- 49a² − 16b⁴ = (7a − 4b²)(7a + 4b²).
Answer: (5x − 3)² and (7a − 4b²)(7a + 4b²)
8. Solve the equation (x + 4)² − (x − 4)² = 32. avanzato
Show solution
- Expand (x + 4)² = x² + 8x + 16.
- Expand (x − 4)² = x² − 8x + 16.
- Subtract, remembering to flip all the signs of the second expansion: x² + 8x + 16 − x² + 8x − 16 = 16x.
- The equation becomes 16x = 32, so x = 2.
- Check: 6² − (−2)² = 36 − 4 = 32.
Answer: x = 2
FAQ
How do I remember the special products formulas?
Remember the structure. A squared binomial always has three terms (first squared, twice the product, second squared), while a sum times a difference has two. The cube follows the coefficients 1, 3, 3, 1 from Pascal's triangle.
Why is (a + b)² not equal to a² + b²?
Because (a + b)² means (a + b)(a + b), and every term multiplies every other term. You get a², ab, ab and b². The two ab terms add up to the middle term 2ab.
What are special products used for?
They let you expand products without multiplying term by term, factor polynomials, simplify algebraic fractions and do mental math such as 99² or 101².
How do I know if a trinomial is a perfect square?
Check that two terms are perfect squares and that the third is twice the product of their roots. If so, the trinomial is the square of a binomial. If not, factor it another way.
What is the difference between a squared binomial and a difference of squares?
A squared binomial such as (a − b)² has three terms and a middle term. A difference of squares, a² − b², has two terms and factors as (a + b)(a − b).