Linear and Quadratic Equations

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is raised only to the power of one. Solving these equations is a fundamental skill in business calculus, as they often model relationships between quantities.

• Definition: A linear equation has the form ax + b = c.

• Steps to Solve:

  1. Isolate the variable on one side of the equation.

  2. Simplify both sides as needed.

• Example: Solve

  • Add 10 to both sides:

  • Divide by 5:

• Example: Solve

  • Expand and simplify:

  • Combine like terms:

  • Add to both sides: (No solution)

Solving Quadratic Equations

Quadratic equations are second-degree equations, commonly encountered in business applications such as profit maximization and cost analysis.

• Definition: A quadratic equation has the form .

• Factoring: If possible, factor the equation to find the roots.

• Quadratic Formula: If factoring is not possible, use the quadratic formula:

• Example (Factorable):

  • Factor:

  • Solutions: ,

• Example (Not Factorable):

  • Use quadratic formula: , ,

  • Solutions: ,

Rational Equations and Domain

Solving Rational Equations

Rational equations involve fractions with polynomials in the numerator and denominator. It is important to consider restrictions on the variable that make the denominator zero.

• Definition: A rational equation is an equation containing at least one rational expression.

• Steps to Solve:

  1. Find a common denominator (LCD).

  2. Multiply both sides by the LCD to clear denominators.

  3. Solve the resulting equation.

  4. Check for extraneous solutions (values that make any denominator zero).

• Example:

  • LCD is

  • Multiply both sides by LCD and solve.

  • Check that , (domain restrictions).

Domain of Functions

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

• Domain Rules:

  • Denominator cannot be zero.

  • Expression under an even root must be non-negative.

  • Argument of a logarithm must be positive.

• Example: For , set so .

• Example: For , set so .

Factoring Techniques

Factoring Polynomials

Factoring is the process of writing a polynomial as a product of its factors. This is essential for solving equations and simplifying expressions in calculus.

• Difference of Squares:

• Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

• "Bottoms Up" Method: Used for factoring trinomials where the leading coefficient is not 1.

• Example:

• Example:

Exponents and Radicals

Exponent Rules

Understanding exponent rules is crucial for simplifying expressions and solving equations in calculus.

Radical Expressions

Radicals are expressions involving roots. They can be rewritten using fractional exponents.

• Square Root:

• Cube Root:

• General:

• Example:

  • Rewrite using exponents:

  • Apply exponent rules and simplify.

Linear Functions and Slope

Slope and Equation of a Line

Linear functions are used to model constant rates of change, such as cost per item or revenue per sale.

• Slope Formula:

• Slope-Intercept Form:

• Point-Slope Form:

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Slopes are negative reciprocals:

• Example: Find the equation of a line passing through with slope :

  • Point-slope form:

  • Simplify:

Additional info:

• These foundational algebraic skills are essential for success in business calculus, where functions, equations, and domains are frequently analyzed.

• Factoring and solving quadratic equations are particularly important for optimization problems in business contexts.