(7a-ab-4c²)² in 3 methods ​

Answer:three methods to expand (7a - ab - 4c^2)^2: Method 1: Using the formula (x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz Let x = 7a, y = -ab, and z = -4c^2. Then: (7a - ab - 4c^2)^2 = (7a)^2 + (-ab)^2 + (-4c^2)^2 + 2(7a)(-ab) + 2(7a)(-4c^2) + 2(-ab)(-4c^2) = 49a^2 + a^2b^2 + 16c^4 - 14a^2b - 56ac^2 + 8abc^2 Method 2: Repeated multiplication (7a - ab - 4c^2)^2 = (7a - ab - 4c^2)(7a - ab - 4c^2) We can use the distributive property (FOIL method extended): First: (7a)(7a) = 49a^2Outer: (7a)(-ab) = -7a^2bInner: (-ab)(7a) = -7a^2bLast: (-ab)(-ab) = a^2b^2 Then, multiply each term by -4c^2: (7a)(-4c^2) = -28ac^2(-ab)(-4c^2) = 4abc^2(-4c^2)(-4c^2) = 16c^4 Combining all terms: 49a^2 - 7a^2b - 7a^2b + a^2b^2 - 28ac^2 + 4abc^2 + 16c^4 = 49a^2 - 14a^2b + a^2b^2 - 28ac^2 + 8abc^2 + 16c^4 Method 3: Using the binomial theorem (with a slight modification) We can rewrite the expression as: (7a - (ab + 4c^2))^2 Now, we can use the binomial theorem (x - y)^2 = x^2 - 2xy + y^2, where x = 7a and y = (ab + 4c^2): (7a)^2 - 2(7a)(ab + 4c^2) + (ab + 4c^2)^2 = 49a^2 - 14a(ab + 4c^2) + (a^2b^2 + 8abc^2 + 16c^4) = 49a^2 - 14a^2b - 56ac^2 + a^2b^2 + 8abc^2 + 16c^4 = 49a^2 - 14a^2b + a^2b^2 - 56ac^2 + 8abc^2 + 16c^4 All three methods yield the same result: 49a^2 - 14a^2b + a^2b^2 - 56ac^2 + 8abc^2 + 16c^4

Answered by Sapphiremoonlight89 | 2025-06-29