Radicals in Algebra Flashcards

Key Terms in Algebra

• Rational Expression: An algebraic expression represented as a ratio of two polynomial expressions.

• Unlike Radicals: Can be transformed into like radicals by multiplying with their conjugate.

• Conjugate: A binomial created by changing the sign of the second term in a binomial expression.

• Radical Equation: An equation containing a variable within a radicand or featuring a variable with a rational exponent.

• Rational Equation: An equation that involves one or more rational expressions.

• Extraneous Solution: A solution that fails to satisfy the original equation after substitution.

• Square Root Function: A function where the independent variable is situated within a square root.

• Radical Expressions: Any expression that includes a radical sign (√).

• Like Radicals: Radical expressions sharing the same index and radicand.

• Rationalize: The process of eliminating irrational numbers from denominators in expressions.

Important Mathematical Concepts

• Quadratic Formula: The solution for quadratic equations is given by ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

• Discriminant: The part of the quadratic formula under the square root, calculated as ( b^2 - 4ac ).

Types of Numbers

• Complex Numbers: Expressed in the form ( a + bi ), where they are imaginary if ( b \neq 0 ) and real if ( b = 0 ).

• Imaginary Numbers: Formulated as ( a + bi ) with ( b \neq 0 ).

• Rational Numbers: Numbers that can be expressed as the quotient of two integers (fractions).

• Irrational Numbers: Numbers that cannot be represented as fractions.

Radicals and Their Properties

• Radicand: The number situated inside a radical symbol.

• Index: Indicates the degree of the root outside the radical (e.g., square root has an index of 2).

Properties of Imaginary Unit ( i )

• Conjugate of ( a + bi ): Formed as ( a - bi ).

• ( i ): Defined as ( \sqrt{-1} ).

• ( i^2 ): Equals -1.

• ( i^3 ): Equals -i.

• ( i^4 ): Equals 1.

Roots and Their Nature

• Discriminant is Negative: Leads to two imaginary conjugate roots.

• Positive Square: Results in two real rational roots.

• Positive but Not a Square: Yields two real irrational roots.