(a-b)² = a²-2ab+b²
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- The distributive property says for numbers A, B, C,
- A × (B + C) = (A × B) + (A × C)
- How does this apply to (a + b) × (a + b)?
- a + b is just a number, so it is valid to treat (a + b) as A, B, or C in the distributive property, because the property *always* works when A, B, C are numbers.
- So let's connect our expression to the distributive property, by setting a and b to terms with A, B, C.
- A = a + b
- B = a
- C = b
- After plugging in these values,
- A × (B + C) = (A × B) + (A × C) becomes
- (a + b) × (a + b) = ((a + b) × a) + ((a + b) × b)
- Since the order of multiplication does not matter, we can rewrite the right side of that equation…
- ( a + b) × (a + b) = (a × (a + b)) + (b × (a + b))
- Now, using the distributive property more simply,
- (a + b) × (a + b) = (a×a + a×b) + (b×a + b×b)
- The parentheses on the right side do not change order of operations, because the order that we add all of the multiplied terms up does not matter, so
- (a + b) × (a + b) = a×a + a×b + b×a + b×b
- Of course, b×a can be rewritten as a×b, so
- (a + b) × (a + b) = a×a + a×b + a×b + b×b
- Now we can combine the two like-terms of a×b
- (a + b) × (a + b) = a×a + 2×a×b + b×b
- Which is more common written as
- (a+b)2=a2+2ab+b2
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