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freshman mathmatics
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EquationsLinear Equations: An equation of the form (ax + b = 0), where (a) and (b) are constants. The solution is
(x = -\frac{b}{a}).
Quadratic Equations: An equation of the form (ax^2 + bx + c = 0). Solutions can be found using the
quadratic formula:
x=−b±b2−4ac2ax=2a−b±b2−4ac
Systems of Equations: A set of two or more equations with the same variables. Solutions can be found
using methods like substitution, elimination, or matrix operations.
Inequalities
Linear Inequalities: Similar to linear equations but with inequality signs ((<, \leq, >, \geq)). Solutions
are often represented on a number line.
Quadratic Inequalities: Inequalities involving quadratic expressions. Solutions can be found by solving
the corresponding quadratic equation and testing intervals.
Systems of Inequalities: Multiple inequalities that are solved simultaneously. Solutions are often
represented as regions on a coordinate plane.
Functions
Definition: A function is a relation between a set of inputs and a set of possible outputs where each
input is related to exactly one output.
Types of Functions:
Linear Functions: (f(x) = mx + b)
Quadratic Functions: (f(x) = ax^2 + bx + c)
Polynomial Functions: Functions involving terms with variables raised to whole number
exponents.
Exponential Functions: (f(x) = a \cdot b^x)
Logarithmic Functions: The inverse of exponential functions, (f(x) = \log_b(x))
Properties:
Domain and Range: The set of all possible inputs (domain) and outputs (range).
Intercepts: Points where the function crosses the x-axis (x-intercepts) and y-axis (y-intercepts).
Asymptotes: Lines that the graph of the function approaches but never touches.
Theorems
Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the
squares of the other two sides:
a2+b2=c2a2+b2=c2
Fundamental Theorem of Algebra: Every non-zero polynomial equation of degree (n) has exactly (n)
roots (including complex and repeated roots).
Intermediate Value Theorem: If a continuous function (f) takes on values (f(a)) and (f(b)) at two points
(a) and (b), then it also takes on any value between (f(a)) and (f(b)) at some point within ([a, b]).
Binomial Theorem: Describes the algebraic expansion of powers of a binomial. For any positive
integer (n):
(a+b)n=∑k=0n(nk)an−kbk(a+b)n=k=0∑n(kn)an−kbk
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