Quadratic Equations Guide

A quadratic equation is a second-degree polynomial equation in the form:

$ax^2 + bx + c = 0$

Methods of Solving

1. Factoring

When the equation can be factored:

$$\begin{align*} x^2 + 5x + 6 &= 0 \\ (x + 2)(x + 3) &= 0 \\ x &= -2 \text{ or } x = -3 \end{align*}$$

2. Quadratic Formula

The universal method that always works:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Solve $2x^2 + 7x + 3 = 0$

$$\begin{align*} a &= 2, \quad b = 7, \quad c = 3 \\ x &= \frac{-7 \pm \sqrt{49 - 24}}{4} \\ x &= \frac{-7 \pm 5}{4} \\ x &= -0.5 \text{ or } x = -3 \end{align*}$$

3. Completing the Square

Useful for deriving the quadratic formula:

$$\begin{align*} x^2 + 6x + 5 &= 0 \\ x^2 + 6x &= -5 \\ x^2 + 6x + 9 &= -5 + 9 \\ (x + 3)^2 &= 4 \\ x + 3 &= \pm 2 \\ x &= -1 \text{ or } x = -5 \end{align*}$$

The Discriminant

$\Delta = b^2 - 4ac$

Determines the nature of roots:

• $\Delta > 0$: Two distinct real roots
• $\Delta = 0$: One repeated real root
• $\Delta < 0$: Two complex conjugate roots

Properties of Roots

For equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$:

• Sum of roots: $\alpha + \beta = -\frac{b}{a}$
• Product of roots: $\alpha\beta = \frac{c}{a}$

Applications

1. Projectile motion: Height equations
2. Area problems: Optimization
3. Business: Revenue and profit maximization
4. Engineering: Structural calculations

Graph Characteristics

Parabola:

$y = ax^2 + bx + c$

• Vertex: Minimum/maximum point
• Axis of symmetry: x = -b/2a
• Opens upward if a > 0, downward if a < 0

Understanding quadratics is fundamental for advanced mathematics and many practical applications.