2.9.1: Key Terms

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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: $A = L W$

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 $a + b i,$ 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 $−1$ : $i = \sqrt{−1} .$

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, $(x, y)$

origin: the point where the two axes cross in the center of the plane, described by the ordered pair $(0, 0)$

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: $P = 2 L + 2 W$

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: $V = L W H$

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

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