Simplification Math Example 2

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Example 2

medium
Simplify the algebraic expression x2โˆ’9x2โˆ’xโˆ’6\dfrac{x^2 - 9}{x^2 - x - 6} and state any restrictions.

Solution

  1. 1
    Factor numerator: x2โˆ’9=(xโˆ’3)(x+3)x^2 - 9 = (x-3)(x+3).
  2. 2
    Factor denominator: x2โˆ’xโˆ’6=(xโˆ’3)(x+2)x^2 - x - 6 = (x-3)(x+2).
  3. 3
    Cancel the common factor (xโˆ’3)(x-3), valid when xโ‰ 3x \ne 3: (x+3)(x+2)\dfrac{(x+3)}{(x+2)}.
  4. 4
    Restriction: xโ‰ 3x \ne 3 (original denominator zero) and xโ‰ โˆ’2x \ne -2 (simplified denominator zero).

Answer

x2โˆ’9x2โˆ’xโˆ’6=x+3x+2,xโ‰ 3,โ€…โ€Šxโ‰ โˆ’2\frac{x^2-9}{x^2-x-6} = \frac{x+3}{x+2},\quad x \ne 3,\; x \ne -2
Simplifying rational expressions requires factoring and cancelling common factors, while keeping track of values that make any denominator zero โ€” these are excluded from the domain.

About Simplification

The process of replacing a complex expression or model with a simpler equivalent that preserves the essential features.

Learn more about Simplification โ†’

More Simplification Examples

Example 1 easy

Simplify the Boolean expression [formula] using absorption laws.

Example 3 easy

Simplify: [formula].

Example 4 medium

Simplify the set expression [formula] and justify each step.

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