Quadratic Equation Basics

Quadratic Equation

Definition

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two.

General Form

The general form of a quadratic equation is:

ax^2 + bx + c = 0

where:

• a, b, and c are constants (numbers)
• a ≠ 0 (if a = 0, it's not a quadratic equation)

Solutions

A quadratic equation can have:

• Two distinct real roots (solutions)
• One repeated real root (solution)
• No real roots (complex roots)

Methods for Solving

1. Factoring

• If the equation can be written in the form (x - r)(x - s) = 0, then the roots are x = r and x = s
• Not all quadratic equations can be factored

2. Quadratic Formula

• If the equation cannot be factored, use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

• This formula will give two solutions for the value of x

Graphing

• The graph of a quadratic equation is a parabola that opens:

  • Upward (a > 0)
  • Downward (a < 0)

• The x-intercepts of the graph represent the roots of the equation

Applications

• Projectile motion
• Optimization problems
• Electrical circuits
• Physics and engineering problems

Quadratic Equation

Definition

• A quadratic equation is a polynomial equation of degree two, meaning the highest power of the variable (usually x) is two.

General Form

• The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants (numbers) and a ≠ 0.
• If a = 0, the equation is not a quadratic equation.

Solutions

• A quadratic equation can have two distinct real roots (solutions), one repeated real root (solution), or no real roots (complex roots).

Methods for Solving

Factoring

• If the equation can be written in the form (x - r)(x - s) = 0, then the roots are x = r and x = s.
• Not all quadratic equations can be factored.

Quadratic Formula

• The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
• This formula will give two solutions for the value of x.
• The quadratic formula is used when the equation cannot be factored.

Graphing

• The graph of a quadratic equation is a parabola that opens upward if a > 0 and downward if a < 0.
• The x-intercepts of the graph represent the roots of the equation.

Applications

• Quadratic equations are used to model various real-world scenarios, including:
  • Projectile motion
  • Optimization problems
  • Electrical circuits
  • Physics and engineering problems