The Main Idea

• Definition of Quadratic Equations:
  • Second-degree polynomial equation
  • Standard form: [latex]ax^2 + bx + c = 0[/latex], where [latex]a \neq 0[/latex]
• Zero-Product Property:
  • If [latex]ab = 0[/latex], then [latex]a = 0[/latex] or [latex]b = 0[/latex]
  • Fundamental to solving factored quadratics
• Grouping Method:
  • Used when the leading coefficient is not 1
  • Transforms the quadratic into four terms for easier factoring
  • Key Steps:
    • Find factors of [latex]ac[/latex] that sum to [latex]b[/latex]
    • Rewrite middle term using these factors
    • Group terms and factor by pairs
    • Factor out common binomial

Solving by Factoring: Step-by-Step

1. Arrange the equation in standard form: [latex]ax^2 + bx + c = 0[/latex]
2. Factor the left side of the equation
3. Apply the zero-product property
4. Solve the resulting linear equations
5. Check solutions in the original equation

The Main Idea

• Square Root Property:
  • Used when there’s no linear term in the quadratic equation
  • If [latex]x^2 = k[/latex], then [latex]x = \pm\sqrt{k}[/latex] (where [latex]k \geq 0[/latex])
• Applying Square Root Property:
  • Isolate [latex]x^2[/latex] term
  • Take square root of both sides
  • Remember to use [latex]\pm[/latex] sign
• Pythagorean Theorem:
  • Relates sides of a right triangle: [latex]a^2 + b^2 = c^2[/latex]
  • [latex]a[/latex] and [latex]b[/latex] are legs, [latex]c[/latex] is hypotenuse
  • Leads to quadratic equations when solving for a side

The Main Idea

• Purpose:
  • Transform [latex]ax^2 + bx + c = 0[/latex] into [latex](x - h)^2 = k[/latex]
  • Enables solving quadratics that can’t be easily factored
• Key Steps:
  • Rearrange the equation so that it is in standard form
  • Isolate [latex]x[/latex] terms on one side
  • Add [latex](\frac{b}{2})^2[/latex]
  • Factor perfect square trinomial
  • Apply square root property
• Perfect Square Trinomials:
  • [latex]x^2 + 2px + p^2 = (x + p)^2[/latex]
  • [latex]x^2 - 2px + p^2 = (x - p)^2[/latex]
• Applications:
  • Solving quadratic equations
  • Finding vertex of parabolas
  • Deriving quadratic formula

The Main Idea

• Quadratic Formula:
  • Solves any quadratic equation [latex]ax^2 + bx + c = 0[/latex]
  • Formula: [latex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/latex]