Question: User: Alg. 2(H), Unit 2 Name ____________\nL23: Conceptual Understanding Date ____________ Pd. ___\n\n1) Verify: $ (a + b)^2 = a^2 + 2ab + b^2 $ by expanding the left side.\n\n2) A student claims: "$(x + 3)^2 = x^2 + 9$ because I square each term"\nExpand correctly and explain the error.\n\n3) An architect uses the identity $ a^2 - b^2 = (a - b)(a + b) $ to\ncalculate areas of regions in building designs. A large square garden\nhas side $100$ m. A smaller square fountain has side $20$ m.\n\na) Find the area of the garden minus fountain using $ a^2 - b^2 $\n(difference of squares).\n\nb) Use the factored form $(100 - 20)(100 + 20)$ to show the\nsame answer.
1. **Problem 1: Verify $ (a + b)^2 = a^2 + 2ab + b^2 $ by expanding the left side.\n\n2. **Formula:** The square of a binomial is given by $ (a + b)^2 = (a + b)(a + b) $.\n\n3. **Expand the left side:**\n$$ (a + b)(a + b) = a(a + b) + b(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2 $$\n\n4. **Explanation:** When multiplying, each term in the first binomial multiplies each term in the second. Since $ab$ and $ba$ are the same, they combine to $2ab$.\n\n5. **Final verification:**\n$$ (a + b)^2 = a^2 + 2ab + b^2 $$\n\n---\n\n6. **Problem 2: Student claims $(x + 3)^2 = x^2 + 9$ by squaring each term. Explain the error.\n\n7. **Correct expansion:**\n$$ (x + 3)^2 = (x + 3)(x + 3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9 $$\n\n8. **Error explanation:** The student incorrectly squared each term separately without considering the cross product terms $2 \, \times \, x \, \times \, 3 = 6x$. The binomial square includes these middle terms, so simply squaring each term is wrong.\n\n---\n\n9. **Problem 3a: Find the area of the garden minus fountain using $ a^2 - b^2 $.\n\n10. **Given:** Large square side $a = 100$ m, smaller square side $b = 20$ m.\n\n11. **Calculate areas:**\n$$ a^2 = 100^2 = 10000 \quad \text{and} \quad b^2 = 20^2 = 400 $$\n\n12. **Difference of areas:**\n$$ a^2 - b^2 = 10000 - 400 = 9600 $$\n\n13. **Interpretation:** The area of the garden minus the fountain is $9600$ square meters.\n\n---\n\n14. **Problem 3b: Use factored form $(100 - 20)(100 + 20)$ to show the same answer.\n\n15. **Calculate:**\n$$ (100 - 20)(100 + 20) = 80 \times 120 = 9600 $$\n\n16. **Conclusion:** Both methods give the same area $9600$ square meters, confirming the identity $a^2 - b^2 = (a - b)(a + b)$.