Quadratic equations are powerful tools for solving real-world problems. They model situations like projectile motion and profit optimization. Understanding how to factor and solve these equations is key to unlocking their potential.

The Zero Product Property and factoring techniques are essential for solving quadratic equations. Graphing parabolas helps visualize solutions, while the quadratic formula provides a reliable method when factoring fails. These skills open doors to advanced problem-solving.

Quadratic Equations

Zero Product Property application

• States if the product of factors is zero, then at least one factor must be zero
  • Example: If $ab = 0$, then either $a = 0$, $b = 0$, or both are zero
• Solving quadratic equations using Zero Product Property:
  1. Set quadratic expression equal to zero
  2. Factor the quadratic expression into its component factors
  3. Set each factor equal to zero and solve for the variable
  4. Solutions are values that make each factor equal zero (roots)

Factoring quadratic expressions

• Rewriting polynomial as product of factors
• Factoring quadratic expression $ax^2 + bx + c$:
  • Find two numbers with product $ac$ and sum $b$
  • Rewrite quadratic expression using these numbers
  • Factor by grouping if necessary (splitting middle term)
• After factoring, set each factor to zero and solve for variable to find solutions
  • Quadratic formula can factor when other methods fail: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • The expression under the square root ($b^2 - 4ac$) is called the discriminant

Real-world quadratic modeling

• Quadratic equations model various situations:
  • Height of object thrown upward (projectile motion)
  • Area of rectangular space (optimization)
  • Profit of business (revenue and cost functions)
• Solving real-world problems with quadratic equations:
  1. Identify given information and unknown variable
  2. Create quadratic equation representing the situation
  3. Solve quadratic equation by factoring or Zero Product Property
  4. Interpret solutions in context of real-world problem
     • Determine which solutions are relevant and meaningful for situation
     • Example: Negative time values not applicable in projectile motion

Graphical representation of quadratic equations

• The graph of a quadratic equation forms a parabola
• Key features of a parabola:
  • Vertex: The highest or lowest point of the parabola
  • Axis of symmetry: A vertical line passing through the vertex
  • Roots: The x-intercepts of the parabola (solutions to the quadratic equation)
• Completing the square: A method to rewrite a quadratic equation in vertex form, useful for finding the vertex and axis of symmetry