Quadratic Equations

Definition

A quadratic equation is a second-degree polynomial equation in one variable, equivalent to the standard form: $ax^2 + bx + c = 0 \quad (a \neq 0)$

• How to read: “The A x squared plus b x plus c equals zero, where a is not equal to zero.”
• Meaning / when to use: The canonical form of any quadratic equation. All solving methods (factoring, completing the square, quadratic formula) start from or reduce to this. Use when modeling any situation with a squared term (projectile height, profit curves, area optimization).

Why It Matters

Quadratics are the first step into a non-linear world. If you can’t solve them, you can’t predict the impact point of a projectile, the maximum height of a wave, or the “sweet spot” of a business’s revenue. They allow us to find the specific moments where a system changes its behavior, hits a physical boundary, or reaches peak efficiency.

Core Concepts

• Solving Methods:
  • Factoring: Uses the Zero-Product Property.
  • Square Root Method: Useful for $(x+h)^2 = p$, leading to $x = -h \pm \sqrt{p}$.
  • Completing the Square: Transforming $x^2 + bx$ into a perfect square by adding $(\frac{b}{2})^2$.
  • Quadratic Formula: The universal solution $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• The Discriminant ($D = b^2 - 4ac$):
  • If $D > 0$: Two unequal real solutions.
  • If $D = 0$: One repeated real solution (double root).
  • If $D < 0$: No real solutions (two complex conjugate solutions).

Connected Concepts

• Quadratic Functions
• Quadratic Optimization
• Absolute Value Equations and Absolute Value Inequalities
• Autonomous Differential Equations
• Cauchy-Riemann Equations
• Trade-offs
• Leverage
• Map-Territory Relation
• Inversion
• Bottlenecks