The rule for perfect squares describes numbers that are the result of squaring a whole number. In simpler terms, a perfect square is a number you get when you multiply an integer by itself.
Understanding Perfect Squares
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Definition: A perfect square is an integer that can be expressed as the square of another integer.
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Examples:
- 4 is a perfect square because 2 * 2 = 4 (or 22 = 4).
- 9 is a perfect square because 3 * 3 = 9 (or 32 = 9).
- 16 is a perfect square because 4 * 4 = 16 (or 42 = 16).
Identifying Perfect Squares
To determine if a number is a perfect square, you can try to find its square root. If the square root is an integer (a whole number), then the original number is a perfect square.
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Example 1: Is 25 a perfect square?
- The square root of 25 is 5.
- Since 5 is an integer, 25 is a perfect square.
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Example 2: Is 30 a perfect square?
- The square root of 30 is approximately 5.477.
- Since 5.477 is not an integer, 30 is not a perfect square.
Perfect Square Formulas
In algebra, there are two common formulas involving perfect squares, often referred to as perfect square trinomials:
- (a + b)2 = a2 + 2ab + b2
- (a - b)2 = a2 - 2ab + b2
These formulas provide a rule for expanding the square of a binomial.
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Example using (a + b)2:
Let a = 2 and b = 3
(2 + 3)2 = 22 + 2(2)(3) + 32 = 4 + 12 + 9 = 25
Also, (2 + 3)2 = 52 = 25 -
Example using (a - b)2:
Let a = 5 and b = 2
(5 - 2)2 = 52 - 2(5)(2) + 22 = 25 - 20 + 4 = 9
Also, (5 - 2)2 = 32 = 9
In summary, a perfect square is the result of squaring an integer, and perfect square formulas in algebra provide a pattern for expanding the square of a binomial.
