Discover and apply key algebraic identities, including (a+b)², (a−b)², and (a+b)(a−b), building geometric intuition for each.
The identity (a+b)²=a²+2ab+b² can be visualized as the area of a square with side (a+b), split into four regions: a², ab, ab, and b².
Use the identity to expand (x+4)².
x²+8x+16
Use the identity to expand (y−3)².
y²−6y+9
Identities like (a+b)(a−b)=a²−b² provide shortcuts for multiplying numbers close to round values.
Find 97×103 using an identity.
(100−3)(100+3)=100²−3²=10,000−9=9,991
Find 52² using the identity for (a+b)².
(50+2)²=2500+200+4=2,704
1. Expand (x+7)².
2. Expand (y−5)².
3. Find 998×1002 using an identity.
4. Find 48² using an identity.
5. Expand (2a+3b)(2a−3b).