MATH 1325 - Linear, Quadratic and Rational Equations

Vocabulary and key concepts regarding linear, quadratic, and rational equations as discussed in MATH 1325.

Expression

A number, a variable, or a combination of numbers and variables and operation symbols.

Equation

Made up of two expressions connected by an equal sign.

Linear Equation

An equation of the form $ax + b = c$ with $a \neq 0$.

Quadratic Equation

An equation of the form $ax^2 + bx + c = 0$ with $a \neq 0$, usually written in standard form.

Discriminant (D)

A value calculated as $D = b^2 - 4ac$ to determine the nature of the roots of a quadratic equation.

Real Roots ($D > 0$)

There are two real roots given by x_1 = \frac{-b + \text{\sqrt{D}}}{2a} and x_2 = \frac{-b - \text{\sqrt{D}}}{2a}.

Real Roots ($D = 0$)

There is only one real root with multiplicity 2 given by $x_1 = x_2 = \frac{-b}{2a}$.

Real Roots ($D < 0$)

There are no real roots.

Factorization Method

A method to solve quadratic equations by writing the equation in standard form, factoring the polynomial on the left side, and setting each factor equal to zero.

Fake solutions

Potential solutions for rational equations that must be checked to ensure they do not result in invalid mathematical expressions.

Standard Form (Quadratic)

The representation of a quadratic equation as $ax^2 + bx + c = 0$, where the right side of the equation is zero.