Rewriting Expressions Formula

The Formula

x^2 - a^2 = (x + a)(x - a)

When to use: 2(x + 3) and 2x + 6 look different but are the same—rewriting shows this.

Quick Example

x^2 - 4 = (x+2)(x-2) — same expression in factored form, which shows the zeros at x = 2 and x = -2.

Notation

Equivalent forms connected by =. Common forms: expanded (ax^2 + bx + c), factored ((x + p)(x + q)), and simplified (fewest terms).

What This Formula Means

Transforming an algebraic expression into a different but mathematically equivalent form to reveal new information.

2(x + 3) and 2x + 6 look different but are the same—rewriting shows this.

Formal View

Two expressions E_1(x) and E_2(x) are equivalent iff \forall x \in D:\; E_1(x) = E_2(x), where D is their common domain. Rewriting preserves the function E: D \to \mathbb{R} while changing its syntactic representation.

Worked Examples

Example 1

easy
Rewrite x^2 - 25 in factored form.

Solution

  1. 1
    Recognize as a difference of squares: x^2 - 5^2.
  2. 2
    Apply the pattern: a^2 - b^2 = (a+b)(a-b).
  3. 3
    Result: (x+5)(x-5).

Answer

(x+5)(x-5)
Rewriting an expression means transforming it into an equivalent form. The difference of squares pattern is one of the most useful rewriting tools in algebra.

Example 2

medium
Rewrite \frac{x^2 - 4}{x - 2} in simplified form.

Common Mistakes

  • Rewriting an expression into a different form and accidentally changing its value
  • Distributing or factoring only partially — e.g., writing 2(x + 3) = 2x + 3 instead of 2x + 6
  • Assuming two expressions that look different must have different values

Why This Formula Matters

The 'right' form depends on what you're trying to find or show.

Frequently Asked Questions

What is the Rewriting Expressions formula?

Transforming an algebraic expression into a different but mathematically equivalent form to reveal new information.

How do you use the Rewriting Expressions formula?

2(x + 3) and 2x + 6 look different but are the same—rewriting shows this.

What do the symbols mean in the Rewriting Expressions formula?

Equivalent forms connected by =. Common forms: expanded (ax^2 + bx + c), factored ((x + p)(x + q)), and simplified (fewest terms).

Why is the Rewriting Expressions formula important in Math?

The 'right' form depends on what you're trying to find or show.

What do students get wrong about Rewriting Expressions?

Choosing which form to rewrite into — factored, expanded, or simplified — depends on what the problem is asking.

What should I learn before the Rewriting Expressions formula?

Before studying the Rewriting Expressions formula, you should understand: expressions, distributive property.

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