Algebra: Systems of Equations and Quadratic Equations Quiz

Systems of Equations

• A system of equations is a set of two or more equations that must be true at the same time
• Can be solved using substitution, elimination, or graphical methods
• Types of systems:
  • Independent systems: have a unique solution
  • Dependent systems: have infinitely many solutions
  • Inconsistent systems: have no solution

Quadratic Equations

• A quadratic equation is a polynomial equation of degree two
• Standard form: ax^2 + bx + c = 0, where a, b, and c are constants
• Can be solved using:
  • Factoring
  • Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
• Types of solutions:
  • Real and rational
  • Real and irrational
  • Complex

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range)
• Notation: f(x) = output
• Types of functions:
  • Linear: f(x) = mx + b
  • Quadratic: f(x) = ax^2 + bx + c
  • Exponential: f(x) = a^x
  • Logarithmic: f(x) = loga(x)
• Properties:
  • Domain and range
  • Even and odd functions
  • Composition of functions

Complex Numbers

• A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit (i^2 = -1)
• Operations:
  • Addition and subtraction
  • Multiplication and division
• Properties:
  • Conjugate: a - bi
  • Modulus: |a + bi| = √(a^2 + b^2)
  • Argument: arg(a + bi) = tan^(-1)(b/a)

Inequalities

• An inequality is a statement that one expression is greater than, less than, or equal to another
• Notation:
  • > (greater than)
  • < (less than)
  • ≥ (greater than or equal to)
  • ≤ (less than or equal to)
• Solving inequalities:
  • Add or subtract the same value to both sides
  • Multiply or divide both sides by a positive value
• Graphing inequalities:
  • Shade above or below the line
  • Solid line for ≥ or ≤, dashed line for > or <

Asymptotes

• A vertical asymptote is a vertical line that the graph of a function approaches as x approaches a certain value
• A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches infinity or negative infinity
• Types of asymptotes:
  • Vertical asymptotes: occur at x-values where the function is undefined
  • Horizontal asymptotes: occur as x approaches infinity or negative infinity

Logarithmic Equations

• A logarithmic equation is an equation involving logarithms
• Types of logarithms:
  • Natural logarithm (ln)
  • Common logarithm (log)
• Properties:
  • Logarithmic identity: loga(x) = y ⇔ a^y = x
  • Logarithmic rules:
    • Product rule: loga(xy) = loga(x) + loga(y)
    • Quotient rule: loga(x/y) = loga(x) - loga(y)
    • Power rule: loga(x^y) = y loga(x)

Compound Interest with e

• Compound interest is interest calculated on both the principal and any accrued interest
• Formula: A = P(1 + r/n)^(nt), where A is the amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years
• The number e:
  • Approximately equal to 2.718
  • Used as the base for continuous compounding: A = Pe^(rt)

Systems of Equations

• A system of equations is a set of two or more equations that must be true at the same time
• Can be solved using substitution, elimination, or graphical methods
• Independent systems have a unique solution
• Dependent systems have infinitely many solutions
• Inconsistent systems have no solution

Quadratic Equations

• A quadratic equation is a polynomial equation of degree two
• Standard form: ax^2 + bx + c = 0, where a, b, and c are constants
• Can be solved using factoring and the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
• Types of solutions: real and rational, real and irrational, and complex

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range)
• Notation: f(x) = output
• Types of functions: linear, quadratic, exponential, and logarithmic
• Properties: domain and range, even and odd functions, and composition of functions

Complex Numbers

• A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit (i^2 = -1)
• Operations: addition, subtraction, multiplication, and division
• Properties: conjugate, modulus, and argument
• Conjugate: a - bi
• Modulus: |a + bi| = √(a^2 + b^2)
• Argument: arg(a + bi) = tan^(-1)(b/a)

Inequalities

• An inequality is a statement that one expression is greater than, less than, or equal to another
• Notation: >, <, ≥, and ≤
• Solving inequalities: add or subtract the same value to both sides, and multiply or divide both sides by a positive value
• Graphing inequalities: shade above or below the line, and use solid or dashed lines for different types of inequalities

Asymptotes

• A vertical asymptote is a vertical line that the graph of a function approaches as x approaches a certain value
• A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches infinity or negative infinity
• Types of asymptotes: vertical and horizontal
• Vertical asymptotes occur at x-values where the function is undefined
• Horizontal asymptotes occur as x approaches infinity or negative infinity

Logarithmic Equations

• A logarithmic equation is an equation involving logarithms
• Types of logarithms: natural logarithm (ln) and common logarithm (log)
• Logarithmic identity: loga(x) = y ⇔ a^y = x
• Logarithmic rules: product, quotient, and power rules

Compound Interest with e

• Compound interest is interest calculated on both the principal and any accrued interest
• Formula: A = P(1 + r/n)^(nt), where A is the amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years
• The number e: approximately equal to 2.718, used as the base for continuous compounding: A = Pe^(rt)