Learn about algebraic formulae for expanding squares and how to apply them.
Example 1: Expansion of (a + b)2
Explain and apply the formula: (a + b)2 = a2 + 2ab + b2.
Formula: The square of the sum of two terms is given by:
(a + b)2 = a2 + 2ab + b2
Explanation: (a + b)2 means (a + b) multiplied by (a + b).
(a + b)(a + b) = a(a + b) + b(a + b)
= a*a + a*b + b*a + b*b
= a2 + ab + ab + b2
= a2 + 2ab + b2
Example: Expand (x + 3)2.
Step 1: Compare (x + 3)2 with (a + b)2. Here, a = x and b = 3.
Step 2: Apply the formula (a + b)2 = a2 + 2ab + b2.
(x + 3)2 = x2 + 2(x)(3) + 32
Step 3: Simplify each term.
x2 = x2
2(x)(3) = 6x
32 = 9
Result: Combine the simplified terms.
(x + 3)2 = x2 + 6x + 9.
Example 2: Expansion of (a - b)2
Explain and apply the formula: (a - b)2 = a2 - 2ab + b2.
Formula: The square of the difference of two terms is given by:
(a - b)2 = a2 - 2ab + b2
Explanation: (a - b)2 means (a - b) multiplied by (a - b).
(a - b)(a - b) = a(a - b) - b(a - b)
= a*a - a*b - b*a + b*b
= a2 - ab - ab + b2
= a2 - 2ab + b2
Example: Expand (y - 5)2.
Step 1: Compare (y - 5)2 with (a - b)2. Here, a = y and b = 5.
Step 2: Apply the formula (a - b)2 = a2 - 2ab + b2.
(y - 5)2 = y2 - 2(y)(5) + 52
Step 3: Simplify each term.
y2 = y2
2(y)(5) = 10y
52 = 25
Result: Combine the simplified terms.
(y - 5)2 = y2 - 10y + 25.
Example 3: Expansion with Coefficients
Expand (2m + 3n)2.
Step 1: Compare (2m + 3n)2 with (a + b)2. Here, a = 2m and b = 3n.
Step 2: Apply the formula (a + b)2 = a2 + 2ab + b2.
(2m + 3n)2 = (2m)2 + 2(2m)(3n) + (3n)2
Step 3: Simplify each term. Remember to square both the coefficient and the variable.
(2m)2 = 22 * m2 = 4m2
2(2m)(3n) = 2 * 2 * 3 * m * n = 12mn
(3n)2 = 32 * n2 = 9n2
Result: Combine the simplified terms.
(2m + 3n)2 = 4m2 + 12mn + 9n2.
Example 4: Expansion with Coefficients and Subtraction
Expand (4p - 5q)2.
Step 1: Compare (4p - 5q)2 with (a - b)2. Here, a = 4p and b = 5q.
Step 2: Apply the formula (a - b)2 = a2 - 2ab + b2.
(4p - 5q)2 = (4p)2 - 2(4p)(5q) + (5q)2
Step 3: Simplify each term. Remember to square both the coefficient and the variable.
(4p)2 = 42 * p2 = 16p2
2(4p)(5q) = 2 * 4 * 5 * p * q = 40pq
(5q)2 = 52 * q2 = 25q2
Result: Combine the simplified terms.
(4p - 5q)2 = 16p2 - 40pq + 25q2.
Practice Mode
Enter a simple expression in the form (a+b)^2 or (a-b)^2 to see its expansion.
Note: Enter expressions using formats like "(x+3)^2", "(y-5)^2", "(2m+3n)^2", "(4p-5q)^2". Use standard variables (a-z), numbers, +, -, parentheses(), and ^2 for squared.
Related Concepts
Explore these related mathematical concepts to deepen your understanding of algebraic formulae.
Binomial
An algebraic expression with two terms (e.g., a + b).
Algebraic Identity
An equation that is true for all values of the variables involved.