Direct Proof Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Prove directly: for all real xx, (x−1)(x+1)=x2−1(x-1)(x+1) = x^2 - 1.

Solution

  1. 1
    Let xx be any real number.
  2. 2
    Expand the left side: (x−1)(x+1)=x⋅x+x⋅1+(−1)⋅x+(−1)⋅1(x-1)(x+1) = x \cdot x + x \cdot 1 + (-1) \cdot x + (-1) \cdot 1.
  3. 3
    Simplify: =x2+x−x−1=x2−1= x^2 + x - x - 1 = x^2 - 1.
  4. 4
    The result holds for all x∈Rx \in \mathbb{R}. □\square

Answer

(x−1)(x+1)=x2−1∀x∈R□(x-1)(x+1) = x^2-1 \quad \forall x \in \mathbb{R} \quad \square
A direct algebraic proof expands the expression and simplifies step by step. Each step uses a known algebraic law (distributivity), making the argument airtight.

About Direct Proof

A direct proof establishes a statement P⇒QP \Rightarrow Q by assuming PP is true and using logical steps, definitions, and known theorems to arrive at QQ — the most straightforward proof strategy.

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More Direct Proof Examples

Example 1 easy

Prove directly: the sum of an even integer and an odd integer is odd.

Example 3 easy

Prove directly: if [formula], then [formula].

Example 4 medium

Prove directly: for all integers [formula], [formula] is even.

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