Glossary
Completing the Square
A technique used to solve quadratic equations or to convert them into vertex form by manipulating the equation to create a perfect square trinomial.
Example:
If you have , you can solve it by completing the square, adding 25 to both sides to make the left side .
Compound Interest
Interest calculated on the initial principal and also on the accumulated interest from previous periods, leading to exponential growth of an investment or loan.
Example:
Investing money in a savings account that earns compound interest means your earnings will grow faster over time as interest is added to your principal.
Discriminant
The expression $$b^2 - 4ac$$ from the quadratic formula, which determines the number and type of real solutions a quadratic equation has.
Example:
If the discriminant of a quadratic equation is negative, it means there are no real solutions, which could imply a projectile never reaches a certain height.
Exponent Properties
Rules that govern how exponents behave in mathematical operations, such as the product rule ($$a^m * a^n = a^{m+n}$$) or the power rule ($$(a^m)^n = a^{mn}$$).
Example:
To simplify , you apply exponent properties to multiply the exponents, resulting in .
Exponential Function
A function of the form $$f(x) = a * b^x$$ where the variable is in the exponent, used to model situations with constant percentage growth or decay.
Example:
The rapid spread of a virus can often be described by an exponential function, showing how the number of infected individuals increases over time.
Extraneous Solutions
Solutions derived mathematically from an equation but which do not make sense or are invalid in the context of the original word problem.
Example:
When solving for the width of a garden, a negative value for x would be an extraneous solution because a physical dimension cannot be negative.
Factoring (Quadratic)
A method for solving quadratic equations by rewriting the quadratic expression as a product of two linear factors, then setting each factor to zero.
Example:
To find the x-intercepts of , you can use factoring to get , revealing solutions at x=1 and x=3.
Growth/Decay Factor
In an exponential function $$f(x) = a * b^x$$, 'b' is the factor by which the quantity changes per unit of time; 'b' > 1 indicates growth, while 0 < 'b' < 1 indicates decay.
Example:
A population increasing by 5% annually has a growth factor of 1.05, while a substance decreasing by 20% per hour has a decay factor of 0.80.
Initial Value (Exponential)
In an exponential function $$f(x) = a * b^x$$, 'a' represents the starting amount or quantity when the independent variable (x) is zero.
Example:
If a savings account starts with 500 is the initial value in the exponential model for its growth.
Log Rules
A set of rules that simplify logarithmic expressions and equations, including rules for products, quotients, and powers within logarithms.
Example:
Using the log rules, you can expand into , which simplifies complex logarithmic equations.
Logarithms
The inverse operation to exponentiation, used to find the exponent to which a base must be raised to produce a given number.
Example:
If you want to find out how many years it takes for an investment to double at a certain interest rate, you'll likely use logarithms to solve for the exponent (time).
Quadratic Equation
An equation of the second degree, meaning it contains at least one term that is squared, typically used to model parabolic paths or areas.
Example:
When a diver jumps from a platform, their height over time can be modeled by a quadratic equation, showing a curved trajectory.
Quadratic Formula
A universal formula, $$x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$$, that provides the solutions for any quadratic equation in standard form.
Example:
When a quadratic equation is too complex to factor, like , the Quadratic Formula is your most reliable tool to find its roots.
Radioactive Decay
The process by which an unstable atomic nucleus loses energy by emitting radiation, modeled by an exponential decay function, often characterized by a half-life.
Example:
Scientists use radioactive decay of carbon-14 to determine the age of ancient artifacts, as the amount of carbon-14 decreases predictably over time.
Solutions (Roots/Zeros)
The values of the variable that satisfy a given equation, often representing the x-intercepts of a function's graph.
Example:
For the equation , the solutions are x=3 and x=-3, which are where the parabola crosses the x-axis.
Standard Form of a Quadratic Equation
The general algebraic representation of a quadratic equation, expressed as $$ax^2 + bx + c = 0$$, where 'a' cannot be zero.
Example:
To solve , you first rewrite it into the Standard Form of a Quadratic Equation as .