Intermediate Algebra: Quadratic Equations

Questions and Answers

What are quadratic equations?

Polynomial equations with a degree of 1

Polynomial equations with a degree of 3

Polynomial equations with a degree of 4

Polynomial equations with a degree of 2 (correct)

What are the roots of the quadratic equation?

Solution of the equation, or the value of the variable that makes the equation true, D = b^2 - 4ac

Two imaginary numbers/conjugate complex numbers occur when D < 0.

True

A double root occurs when D = 0.

True Signup and view all the answers

Irrational conjugates occur when D < 0.

False Signup and view all the answers

What are quadratic inequalities?

Polynomial inequality with a degree of 2 Signup and view all the answers

Equation having a variable in an absolute value sign is called an _________________.

absolute value equation Signup and view all the answers

What does Theorem 1 of Absolute Inequalities state if a >/= 0?

|x| < a then solution is -a < x < a; |x| > a then solution is x > a or x < -a Signup and view all the answers

What does Theorem 2 of Absolute Inequalities state if a < 0?

|x| < a then solution is {}; |x| > a then solution is R Signup and view all the answers

What are rational equations?

Equation whose numerator and denominator are polynomials with integer or rational coefficients Signup and view all the answers

What is the domain of a rational equation?

Is any real number except for the values that make the denominator 0 Signup and view all the answers

What is an extraneous solution?

A solution obtained that does not satisfy the original equation Signup and view all the answers

An expression with a radical sign or fractional exponents is known as a _________________.

radical expression Signup and view all the answers

What are similar radicals?

Radicals with the same integer and same radicand Signup and view all the answers

What is rationalization?

Process of removing radicals from the numerator or denominator of a fraction by multiplying a rationalizing factor with is equal to one Signup and view all the answers

What is the domain of an irrational equation?

Intersection of the domains of all radical expressions in the equation Signup and view all the answers

If the index of the domain of an irrational equation is even, then the radicand should be _________________.

nonnegative Signup and view all the answers

If the index of the domain of an irrational equation is odd, then the radicand could be _________________.

any value Signup and view all the answers

Two lines are parallel if they have _________________.

same slope, different y-intercept Signup and view all the answers

The product of the slope is equal to -1. Two lines are perpendicular if they have _________________.

negative reciprocal of the other line Signup and view all the answers

What are systems of linear equations?

Sets of two or more linear equations involving the same variables Signup and view all the answers

Match the types of systems with their definitions:

Inconsistent = m1 = m2 and b1 is not equal to b2 - Parallel Lines Consistent Independent = m1 is not equal to m2 - One solution, Intersecting Lines Consistent Dependent = m1 = m2 and b1 = b2 - Infinite solutions, Overlapping or coinciding lines Signup and view all the answers

Study Notes

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).

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Explore the key concepts of quadratic equations through flashcards that cover definitions, roots, and the nature of roots. This quiz is designed to enhance your understanding of their properties, including real and imaginary solutions. Perfect for students looking to solidify their grasp on algebra.