Simplifying with Quadratic Equations

Learn how to simplify rational algebraic fractions by factoring quadratic polynomials. Discover the difference between a **removable discontinuity (hole)** and an **infinite discontinuity (vertical asymptote)**.

Interactive Sandbox

Rational Function Plotter

Load presets or adjust the roots of the numerator and denominator. Watch where holes form (common roots that cancel) and where asymptotes sit (roots remaining in the denominator).

Plot: y = N(x) / D(x)

Active Rational Function

Numerator Roots

Root 1 (n₁) 2.0
Root 2 (n₂) 3.0

Denominator Roots

Root 1 (d₁) 2.0
Root 2 (d₂) -2.0
A purple circle indicates a hole (removable point), and a dashed red line represents a vertical asymptote.
Algebraic Solver

Step-by-Step Factoring & Simplification

Trace how factoring the trinomials reveals common factors that cancel, and how we classify the remaining constraints.

Algebraic Simplification


Discontinuities Analysis


Removable Discontinuities (Holes)

None.

Vertical Asymptotes

None.

Evaluation

Conceptual Quiz

Test your understanding of simplifying rational expressions and identifying holes/asymptotes.

Question 1 of 5 Score: 0/0

If the rational function f(x) = (x-3)(x+1) / (x-3)(x-2) is simplified, what exists at x = 3?

Correct Answer!

Explanation text goes here...

Rational Rules Summary


  • Factor trinomials to find linear root terms.
  • Hole: Forms at x = a if (x-a) cancels out.
  • Asymptote: Forms at x = a if (x-a) remains in denominator.
  • Hole values: Evaluate simplified form at x = hole.