Derive the formula "(a+b)² + (a−b)² = 2(a²+b²)"

To derive the formula \((a+b)^2 + (a-b)^2 = 2(a^2 + b^2)\), we'll expand both squares and simplify the expression.

1. Expand \((a+b)^2\):
\[
(a+b)^2 = a^2 + 2ab + b^2
\]

2. Expand \((a-b)^2\):
\[
(a-b)^2 = a^2 - 2ab + b^2
\]

3. Add the two expanded forms together:
\[
(a+b)^2 + (a-b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2)
\]

4. Combine like terms:
\[
(a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = a^2 + a^2 + b^2 + b^2 + 2ab - 2ab
\]

5. Simplify the combined terms:
\[
2a^2 + 2b^2
\]

6. Factor out the 2:
\[
2(a^2 + b^2)
\]

Thus, we have derived that:
\[
(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)
\]

Would you like any further details or have any questions about this derivation?

Here are 5 numbered questions you might ask next:
1. How do you derive the formula for \((a+b+c)^2\)?
2. Can you explain the Binomial Theorem?
3. How do you prove the Pythagorean Theorem?
4. Can you solve and explain a quadratic equation?
5. How do you find the derivative of a function?

Tip: When expanding algebraic expressions, carefully apply the distributive property and combine like terms for accurate results.

Learn how to derive the formula (a+b)^2 + (a-b)^2 = 2(a^2 + b^2) with a step-by-step explanation. This problem covers key algebraic concepts, including the binomial theorem and algebraic simplification. Suitable for students in Grades 7-9.

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