Key Terms

Key Terms

absolute value equation

an equation in which the variable appears in absolute value bars, typically with two solutions, one accounting for the positive expression and one for the negative expression

area

in square units, the area formula used in this section is used to find the area of any two-dimensional rectangular region: $A=LW$

Cartesian coordinate system

a grid system designed with perpendicular axes invented by René Descartes

completing the square

a process for solving quadratic equations in which terms are added to or subtracted from both sides of the equation in order to make one side a perfect square

complex conjugate

a complex number containing the same terms as another complex number, but with the opposite operator. Multiplying a complex number by its conjugate yields a real number.

complex number

the sum of a real number and an imaginary number; the standard form is $a+bi,$ where a is the real part and $b$ is the complex part.

complex plane

the coordinate plane in which the horizontal axis represents the real component of a complex number, and the vertical axis represents the imaginary component, labeled i.

compound inequality

a problem or a statement that includes two inequalities

conditional equation

an equation that is true for some values of the variable

discriminant

the expression under the radical in the quadratic formula that indicates the nature of the solutions, real or complex, rational or irrational, single or double roots.

distance formula

a formula that can be used to find the length of a line segment if the endpoints are known

equation in two variables

a mathematical statement, typically written in x and y, in which two expressions are equal

equations in quadratic form

equations with a power other than 2 but with a middle term with an exponent that is one-half the exponent of the leading term

extraneous solutions

any solutions obtained that are not valid in the original equation

graph in two variables

the graph of an equation in two variables, which is always shown in two variables in the two-dimensional plane

identity equation

an equation that is true for all values of the variable

imaginary number

the square root of $\mathrm{-1}$ : $i=\sqrt{\mathrm{-1}}.$

inconsistent equation

an equation producing a false result

intercepts

the points at which the graph of an equation crosses the x-axis and the y-axis

interval

an interval describes a set of numbers within which a solution falls

interval notation

a mathematical statement that describes a solution set and uses parentheses or brackets to indicate where an interval begins and ends

linear equation

an algebraic equation in which each term is either a constant or the product of a constant and the first power of a variable

linear inequality

similar to a linear equation except that the solutions will include sets of numbers

midpoint formula

a formula to find the point that divides a line segment into two parts of equal length

ordered pair

a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also known as a coordinate pair, $\left(x,y\right)$

origin

the point where the two axes cross in the center of the plane, described by the ordered pair $\left(0,0\right)$

perimeter

in linear units, the perimeter formula is used to find the linear measurement, or outside length and width, around a two-dimensional regular object; for a rectangle: $P=2L+2W$

polynomial equation

an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents

Pythagorean Theorem

a theorem that states the relationship among the lengths of the sides of a right triangle, used to solve right triangle problems

quadrant

one quarter of the coordinate plane, created when the axes divide the plane into four sections

quadratic equation

an equation containing a second-degree polynomial; can be solved using multiple methods

quadratic formula

a formula that will solve all quadratic equations

radical equation

an equation containing at least one radical term where the variable is part of the radicand

rational equation

an equation consisting of a fraction of polynomials

slope

the change in y-values over the change in x-values

solution set

the set of all solutions to an equation

square root property

one of the methods used to solve a quadratic equation, in which the $x^2$ term is isolated so that the square root of both sides of the equation can be taken to solve for x

volume

in cubic units, the volume measurement includes length, width, and depth: $V=LWH$

x-axis

the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right

x-coordinate

the first coordinate of an ordered pair, representing the horizontal displacement and direction from the origin

x-intercept

the point where a graph intersects the x-axis; an ordered pair with a y-coordinate of zero

y-axis

the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top

y-coordinate

the second coordinate of an ordered pair, representing the vertical displacement and direction from the origin

y-intercept

a point where a graph intercepts the y-axis; an ordered pair with an x-coordinate of zero

zero-product property

the property that formally states that multiplication by zero is zero, so that each factor of a quadratic equation can be set equal to zero to solve equations