Quadratic Equation and Discriminant Analysis

Quadratic Equation

• A quadratic equation is a second-degree polynomial equation in the standard form:
  • ( ax^2 + bx + c = 0 )
  • where ( a \neq 0 ).

Discriminant Analysis

• The discriminant (( D )) is given by:

  • ( D = b^2 - 4ac )

• It indicates the nature of the roots of the quadratic equation:

  1. Real and Distinct Roots:

     • If ( D > 0 ):
       • Two distinct real roots exist.
       • Roots can be calculated using:
         • ( x = \frac{-b \pm \sqrt{D}}{2a} )

  2. Real and Repeated Roots:

     • If ( D = 0 ):
       • One real root with multiplicity two (repeated).
       • Root is:
         • ( x = \frac{-b}{2a} )

  3. Complex Roots:

     • If ( D < 0 ):
       • Two complex conjugate roots exist.
       • Roots can be expressed as:
         • ( x = \frac{-b \pm i\sqrt{|D|}}{2a} )

Real and Complex Roots

• Real Roots:

  • Occur when the discriminant is non-negative (( D \geq 0 )).
  • Can be distinct or repeated based on the value of ( D ).

• Complex Roots:

  • Occur when the discriminant is negative (( D < 0 )).
  • Roots are non-real and come in conjugate pairs, represented as:
    • ( x_1 = p + qi )
    • ( x_2 = p - qi )
  • ( p ) and ( q ) are real numbers, and ( i ) is the imaginary unit.

Summary

• The discriminant helps determine the nature of roots in quadratic equations.
• Real roots depend on a non-negative discriminant, while complex roots arise from a negative discriminant.

Quadratic Equation

• A quadratic equation is represented in the form ( ax^2 + bx + c = 0 ) where ( a ) is not equal to zero.
• It is classified as a second-degree polynomial equation.

Discriminant Analysis

• The discriminant ( D ) is calculated using the formula ( D = b^2 - 4ac ).
• The value of the discriminant determines the nature of the roots:
• Real and Distinct Roots*:
  • Occur when ( D > 0 ), yielding two different real roots.
  • Those roots can be found with the quadratic formula: ( x = \frac{-b \pm \sqrt{D}}{2a} ).
• Real and Repeated Roots*:
  • Occur when ( D = 0 ), resulting in one real root that has a multiplicity of two.
  • The repeated root can be calculated as: ( x = \frac{-b}{2a} ).
• Complex Roots*:
  • Occur when ( D < 0 ), leading to two complex conjugate roots.
  • These roots can be expressed as: ( x = \frac{-b \pm i\sqrt{|D|}}{2a} ), where ( i ) denotes the imaginary unit.

Real and Complex Roots

• Real Roots:

  • Present when the discriminant is non-negative (( D \geq 0 )).
  • Can be either distinct (two different roots) or repeated (one root with multiplicity).

• Complex Roots:

  • Arise when the discriminant is negative (( D < 0 )).
  • Such roots are non-real and appear in conjugate pairs, formatted as:
    • ( x_1 = p + qi )
    • ( x_2 = p - qi )
  • In the above, ( p ) and ( q ) are real numbers, and ( i ) represents the imaginary unit.

Summary

• The discriminant is a key component in identifying the type of roots in quadratic equations.
• Non-negative discriminants lead to real roots, while negative discriminants indicate the presence of complex roots.

Quadratic Equation

• A polynomial equation of degree 2, typically in the form ( ax^2 + bx + c = 0 ).
• Constants ( a ), ( b ), and ( c ) are included, where ( a \neq 0 ).

Graphical Representation

• The graph is a parabola.
• Opens upwards if ( a > 0 ) and downwards if ( a < 0 ).
• Vertex: The point where the parabola achieves its maximum or minimum.
• Axis of Symmetry: A vertical line through the vertex, defined by ( x = -\frac{b}{2a} ).
• Y-Intercept: Found at ( x = 0 ), equal to ( c ).
• X-Intercepts: Solutions to ( ax^2 + bx + c = 0 ) representing real roots.

Vertex Form

• The equation in vertex form is ( y = a(x - h)^2 + k ), where ( (h, k) ) is the vertex.
• Quadratics in standard form can be converted to vertex form using the method of completing the square.

Real and Complex Roots

• Roots are the solutions to the quadratic equation.
• Real Roots: Present when the parabola crosses the x-axis.
• Complex Roots: Occur when the parabola does not intersect the x-axis.
• Root Calculation: Derived using the quadratic formula:
  • ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

Discriminant Analysis

• The discriminant ( D ) is calculated as ( D = b^2 - 4ac ).
• If ( D > 0 ): There are two distinct real roots.
• If ( D = 0 ): There is one real root (repeated).
• If ( D < 0 ): Two complex roots (no real solutions).

Factoring Techniques

• Simple Factorization: Expressing the quadratic as ( (px + q)(rx + s) = 0 ) when possible.
• Using Roots: Factoring as ( a(x - r_1)(x - r_2) = 0 ) when roots ( r_1 ) and ( r_2 ) are known.
• Factoring by Grouping: Applicable when expression can be grouped into pairs.
• Not all quadratics can be easily factored, especially with larger coefficients or irrational roots.