Intermediate Algebra: Quadratic Equations

Quadratic Equations

• Defined as polynomial equations with a degree of 2.
• Fundamental to algebraic solutions and applications.

Roots of the Quadratic Equation

• Solutions are the values of the variable that satisfy the equation.
• The discriminant (D) is calculated as D = b² - 4ac.

Types of Roots

• If D < 0, the equation has two imaginary numbers (conjugate complex numbers).
• If D = 0, there is a double root, meaning real and equal solutions.
• If D > 0, there are irrational conjugates, which are real and unequal solutions.

Quadratic Inequalities

• Inequalities that involve polynomial expressions of degree 2.
• Useful for understanding ranges of values satisfying certain conditions.

Absolute Value Equations

• Contain a variable within an absolute value sign.
• The theorem on absolute inequalities outlines solution ranges based on conditions of 'a':
  • If a ≥ 0, |x| < a results in -a < x < a; |x| > a results in x > a or x < -a.
  • If a < 0, |x| < a yields no solution, while |x| > a results in all real numbers.

Rational Equations / Fractional Equations

• Comprise equations where both the numerator and denominator are polynomials with integer or rational coefficients.

Domain of Rational Equations

• All real numbers are permissible except those that make the denominator zero.

Extraneous Solutions

• Solutions derived from manipulating equations that do not hold true in the original equation context.

Radical Expressions

• Expressions that include radical signs or fractional exponents.

Similar Radicals

• Radicals share the same integer and radicand, allowing for easy combination or simplification.

Rationalization

• Involves removing radicals from the numerator or denominator of a fraction through multiplication by a rationalizing factor (equal to one).

Domain of Irrational Equations

• Defined by the intersection of all radical expressions' domains within the equation.

Nonnegative Radicands

• Required when the index of the domain in an irrational equation is even.

General Radicands

• Any value is permissible when the index is odd in the domain of an irrational equation.

Parallel Lines

• Two lines are parallel if they share the same slope but different y-intercepts.

Slope Relationships

• The slopes of two lines are negative reciprocals if the product equals -1.

Systems of Linear Equations

• Composed of two or more linear equations using identical variables.

Types of Systems

• Inconsistent: Parallel lines (m₁ = m₂ and b₁ ≠ b₂).
• Consistent Independent: One solution with intersecting lines (m₁ ≠ m₂).
• Consistent Dependent: Infinite solutions with overlapping lines (m₁ = m₂ and b₁ = b₂).