Here are some common algebraic proofs across various topics, presented step-by-step:

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1. Proof: The Difference of Squares Formula

Claim: $a^2 - b^2 = (a - b)(a + b)$

Proof:

1. Start with the right-hand side: $(a - b)(a + b)$
2. Expand using the distributive property: $a \cdot a + a \cdot b - b \cdot a - b \cdot b$
3. Simplify the terms: $a^2 + ab - ab - b^2$
4. Combine like terms: $a^2 - b^2$
5. Thus, $a^2 - b^2 = (a - b)(a + b)$.

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2. Proof: Sum of First $n$ Integers

Claim: The sum of the first $n$ integers is given by $S = \frac{n(n+1)}{2}$.

Proof:

1. Write the sum forward and backward: $S = 1 + 2 + 3 + \ldots + n$ $S = n + (n-1) + (n-2) + \ldots + 1$
2. Add the two equations term by term: $2S = (1 + n) + (2 + (n-1)) + \ldots + (n + 1)$
3. Note that each pair sums to $n+1$, and there are $n$ terms: $2S = n(n+1)$
4. Divide by 2: $S = \frac{n(n+1)}{2}$

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3. Proof: $(x + y)^2 = x^2 + 2xy + y^2$

Claim: The square of a binomial expands as given.

Proof:

1. Start with $(x + y)^2$: $(x + y)(x + y)$
2. Apply the distributive property: $x \cdot x + x \cdot y + y \cdot x + y \cdot y$
3. Simplify: $x^2 + xy + xy + y^2$
4. Combine like terms: $x^2 + 2xy + y^2$

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4. Proof: Transitive Property

Claim: If $a = b$ and $b = c$, then $a = c$.

Proof:

1. Start with the first statement: $a = b$
2. Substitute $b = c$ from the second statement into the first: $a = c$
3. This proves the transitive property.

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5. Proof: $x^0 = 1$ (for $x \neq 0$)

Claim: Any non-zero number raised to the power 0 equals 1.

Proof:

1. Start with the property of exponents: $x^m \cdot x^n = x^{m+n}$
2. Set $m = 0$ and $n = k$: $x^0 \cdot x^k = x^{0+k}$
3. Simplify: $x^0 \cdot x^k = x^k$
4. Divide both sides by $x^k$ (valid because $x \neq 0$): $x^0 = 1$

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Do you have any specific type of proof in mind or would you like detailed steps for one of the above?

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Relative Questions:

1. Can you demonstrate the proof for the quadratic formula derivation?
2. How can you prove the sum of a geometric series?
3. Can you prove the $(x-y)^2 = x^2 - 2xy + y^2$ identity?
4. How would you algebraically prove $\sqrt{a^2} = |a|$?
5. What is the proof for the arithmetic mean-geometric mean inequality?

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Tip: Always start with what is given (assumptions or known equations) and manipulate step-by-step towards what needs to be proved.