The Classic Binomial Blunder
Avoid the classic binomial squaring error in Grade 9 algebra: (x-5)² does NOT equal x²+25—always include the middle term using FOIL or the formula (a-b)²=a²-2ab+b².
Key Concepts
Property $(x 5)^2 \neq x^2 + 25$ Remember to either use the pattern for squaring binomials or use the FOIL method.
Explanation Beware the most common trap in algebra! You can't just distribute an exponent over addition or subtraction. Squaring a binomial means multiplying the whole thing by itself, which always creates a middle term. Forgetting that middle term is like baking a cake and forgetting the sugar—the result is just plain wrong. Always use the pattern or FOIL!
Examples Incorrect: $(9x+8)^2 = 81x^2+64$ is false! Correct using FOIL: $(x 5)^2 = (x 5)(x 5) = x^2 5x 5x + 25 = x^2 10x + 25$ Correct using the pattern: $(x 5)^2 = x^2 2(x)(5) + 5^2 = x^2 10x + 25$.
Common Questions
What is the classic binomial squaring blunder?
The most common error is writing (x + a)² = x² + a², completely omitting the middle term. The correct expansion is (x + a)² = x² + 2ax + a². For (x - 5)²: the correct answer is x² - 10x + 25, not x² + 25.
How can you avoid the binomial squaring mistake?
Always write out FOIL when squaring a binomial, or explicitly use the formula (a ± b)² = a² ± 2ab + b². The middle term 2ab is the one students forget. Never skip steps when squaring binomials.
Why does (a + b)² NOT equal a² + b²?
(a + b)² means (a + b)(a + b), not a² + b². FOIL gives a² + ab + ab + b² = a² + 2ab + b². The two cross-product terms ab + ab = 2ab are real and cannot be ignored. Only when a or b is zero does the middle term vanish.