Quadratic Equations and Solving Methods

Quadratic Equation

• Definition: A quadratic equation is a polynomial equation of the form ( ax^2 + bx + c = 0 ), where ( a \neq 0 ).

Solving Methods

1. Factoring:

   • Express the quadratic in the form ( (px + q)(rx + s) = 0 ).
   • Set each factor to zero: ( px + q = 0 ) and ( rx + s = 0 ).
   • Solve for ( x ).

2. Quadratic Formula:

   • Use the formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
   • Discriminant ( D = b^2 - 4ac ):
     • If ( D > 0 ): 2 real and distinct solutions.
     • If ( D = 0 ): 1 real solution (double root).
     • If ( D < 0 ): No real solutions (2 complex solutions).

3. Completing the Square:

   • Rearrange the equation to isolate ( x^2 ): ( ax^2 + bx = -c ).
   • Divide by ( a ) if ( a \neq 1 ): ( x^2 + \frac{b}{a}x = -\frac{c}{a} ).
   • Add ( \left(\frac{b}{2a}\right)^2 ) to both sides.
   • Rewrite the left side as a square: ( \left(x + \frac{b}{2a}\right)^2 = \text{(result)} ).
   • Solve for ( x ) by taking the square root and rearranging.

4. Graphing:

   • Plot the quadratic function ( y = ax^2 + bx + c ).
   • Identify the x-intercepts (roots) where ( y = 0 ).
   • Use vertex form for easier graphing when needed.

5. Numerical Methods (for complex or non-factorable quadratics):

   • Use methods like Newton's method or bisection method to approximate the roots.
   • Useful when analytical methods are difficult or impossible.

Key Notes

• Ensure ( a \neq 0 ) as this defines the equation as quadratic.
• The nature of the roots can be quickly assessed using the discriminant.
• Each method has its advantages depending on the specific equation and context.

Quadratic Equation

• A quadratic equation is expressed as ( ax^2 + bx + c = 0 ), with the condition that ( a \neq 0 ).

Solving Methods

• Factoring:

  • Rewrite the quadratic as ( (px + q)(rx + s) = 0 ).
  • Set each factor to zero: ( px + q = 0 ) and ( rx + s = 0 ) to find values for ( x ).

• Quadratic Formula:

  • The roots can be found using ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
  • The discriminant, ( D = b^2 - 4ac ), determines the nature of the solutions:
    • ( D > 0 ): Two distinct real solutions.
    • ( D = 0 ): One double real solution.
    • ( D < 0 ): Two complex solutions with no real roots.

• Completing the Square:

  • Rearrange to isolate ( x^2 ): ( ax^2 + bx = -c ).
  • If ( a \neq 1 ), divide by ( a ) to normalize.
  • Add ( \left(\frac{b}{2a}\right)^2 ) to both sides for completion.
  • The left side can be rewritten as a square: ( \left(x + \frac{b}{2a}\right)^2 = \text{(result)} ).
  • Solve for ( x ) by taking the square root and rearranging.

• Graphing:

  • Visual representation of the quadratic function ( y = ax^2 + bx + c ).
  • Identify x-intercepts (roots) where the graph intersects the x-axis (where ( y = 0 )).
  • Use vertex form for more straightforward graphing if needed.

• Numerical Methods:

  • Applies to complex or non-factorable quadratics.
  • Techniques like Newton's method or bisection method can be used for root approximation.
  • Particularly useful when analytical solutions are challenging to obtain.

Key Notes

• ( a ) must not equal 0 in order for the equation to be classified as quadratic.
• The discriminant provides a quick assessment of the roots' nature.
• The choice of solution method depends on the specific equation and its characteristics.