Expanding squares such as (x + a)² and (x − a)² appears very often. Once you remember the patterns, you can expand them instantly.
Formula ②: (x + a)² = x² + 2ax + a²
Formula ③: (x − a)² = x² − 2ax + a²
How to remember: “The square of the first term ＋（or −）twice the product of both terms ＋ the square of the last term.” The sign of the middle term is the same as the sign in the original expression.
- (x + 3)² = x² + 2×3×x + 3² = x² + 6x + 9
- (x − 5)² = x² − 2×5×x + 5² = x² − 10x + 25
- (2x + 7)² = (2x)² + 2×7×2x + 7² = 4x² + 28x + 49
Example ①
Expand the following.
(1) (x + 8)² (2) (3x − 4)²
Answer:
(1) x² + 2×8×x + 8² = x² + 16x + 64
(2) (3x)² − 2×4×3x + 4² = 9x² − 24x + 16
Expand the following.
(1) (x + 8)² (2) (3x − 4)²
Answer:
(1) x² + 2×8×x + 8² = x² + 16x + 64
(2) (3x)² − 2×4×3x + 4² = 9x² − 24x + 16
Be careful of common mistakes
(x + a)² ≠ x² + a². Do not forget the middle term 2ax.
For example, (x + 3)² is not x² + 9; it is x² + 6x + 9.
Also, when squaring a variable with a coefficient, square the coefficient as well. (2x)² = 4x²（2² × x²）.
Example ②
Expand (a − 5b)².
Answer:a² − 2×5b×a + (5b)² = a² − 10ab + 25b²
Expand (a − 5b)².
Answer:a² − 2×5b×a + (5b)² = a² − 10ab + 25b²
Key points
- (x+a)² = x² + 2ax + a²、(x−a)² = x² − 2ax + a²
- Never forget the middle term “2ax”
- When squaring, square the entire term, including its coefficient and variables
- The forms (a+b)² and (a−b)² use the same formulas
Practice Problems
- Expand (x + 7)².
- Expand (2x − 3)².
- Expand (4a + 5b)².
Answers and Explanations
- Answer:x² + 14x + 49
Explanation: x² + 2×7×x + 49 = x² + 14x + 49。 - Answer:4x² − 12x + 9
Explanation: (2x)² − 2×3×2x + 9 = 4x² − 12x + 9。 - Answer:16a² + 40ab + 25b²
Explanation: (4a)² + 2×4a×5b + (5b)² = 16a² + 40ab + 25b²。