College Algebra: Essential Formulas and Concepts for Business Applications

Review of Algebra

Business Terms and Formulas

Understanding basic business-related algebraic terms and formulas is essential for solving applied problems in college algebra, especially in business contexts.

• x: Number of units produced or sold.

• p: Price per unit.

• Demand Function: $p = f(x)$, gives the price required to sell x units.

• Total Revenue (R): $R = x p$

• Total Cost (C): $C = \text{Variable Cost} + \text{Fixed Cost}$

• Average Cost per Unit: $C_{\text{avg}} = \frac{C}{x}$

• Total Profit (P): $P = R - C$

• Equilibrium Point: Occurs when supply equals demand, at $(x_0, p_0)$.

Example: If the demand function is $p = 20 - 0.5x$, and the cost function is $C = 100 + 5x$, find the revenue and profit when 10 units are sold.

• Revenue: $R = 10 \times (20 - 0.5 \times 10) = 10 \times 15 = 150$

• Cost: $C = 100 + 5 \times 10 = 150$

• Profit: $P = 150 - 150 = 0$

Review of Algebra

Properties of Exponents

Exponent rules are fundamental for simplifying algebraic expressions and solving equations.

• Negative Exponent: $x^{-n} = \frac{1}{x^n}$

• Product of Powers: $x^m x^n = x^{m+n}$

• Power of a Power: $(x^n)^m = x^{nm}$

• Quotient of Powers: $\frac{x^n}{x^m} = x^{n-m}$

• Power of a Product: $(xy)^n = x^n y^n$

• Power of a Quotient: $\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$

• Zero Exponent: $x^0 = 1$ (for $x \neq 0$)

Example: Simplify $\frac{2^5 \cdot 2^{-3}}{2^2}$.

• $2^5 \cdot 2^{-3} = 2^{5 + (-3)} = 2^2$

• $\frac{2^2}{2^2} = 2^{2-2} = 2^0 = 1$

Review of Algebra

Factorization Techniques

Factoring polynomials is a key skill for simplifying expressions and solving equations.

• Difference of Squares: $x^2 - a^2 = (x - a)(x + a)$

• Difference of Fourth Powers: $x^4 - a^4 = (x^2 - a^2)(x^2 + a^2) = (x - a)(x + a)(x^2 + a^2)$

• Perfect Square Trinomial: $x^2 \pm 2ax + a^2 = (x \pm a)^2$

• Factoring by Grouping: $acx^2 + adx + bcx + bd = (ax + b)(cx + d)$

Example: Factor $x^2 - 9$.

• $x^2 - 9 = (x - 3)(x + 3)$

Equations & Inequalities

Square Root Property

The square root property is used to solve quadratic equations of the form $a^2 = b$.

• If $a^2 = b$, then $a = \pm \sqrt{b}$

Example: Solve $x^2 = 16$.

• $x = \pm 4$

Quadratic Formula

The quadratic formula provides the solutions to any quadratic equation $ax^2 + bx + c = 0$.

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Solve $2x^2 + 3x - 2 = 0$.

• $a = 2, b = 3, c = -2$

• $x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2} = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm 5}{4}$

• So, $x = \frac{2}{4} = 0.5$ or $x = \frac{-8}{4} = -2$

Review of Algebra

Operations with Fractions

Mastering operations with fractions is essential for simplifying algebraic expressions.

• Addition/Subtraction: $\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$

• Multiplication: $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$

• Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}$

Example: $\frac{2}{3} + \frac{1}{4} = \frac{2 \times 4 + 1 \times 3}{3 \times 4} = \frac{8 + 3}{12} = \frac{11}{12}$

Rationalization Techniques

Rationalizing the denominator or numerator involves removing radicals from the denominator or numerator of a fraction.

• If the denominator is $\sqrt{a}$, multiply numerator and denominator by $\sqrt{a}$.

• If the denominator is $a + \sqrt{b}$, multiply numerator and denominator by the conjugate $a - \sqrt{b}$.

Example: Rationalize $\frac{1}{\sqrt{2}}$:

• $\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Graphs of Equations

Slope of a Line

The slope of a line measures its steepness and is calculated using two points on the line.

• Given points $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Example: Find the slope between $(1, 2)$ and $(3, 8)$:

• $m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$

Point-Slope Form of a Line

The point-slope form is useful for writing the equation of a line when you know a point and the slope.

• Equation: $y - y_1 = m(x - x_1)$

Example: Write the equation of a line with slope 2 passing through $(1, 3)$:

• $y - 3 = 2(x - 1)$

Functions

Evaluating Functions

To evaluate a function, substitute the given input value into the function's formula.

• If $f(x) = 2x + 1$, then $f(3) = 2 \times 3 + 1 = 7$

Domain of Functions

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

• For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x = 2$.

• For $f(x) = \sqrt{x}$, the domain is $x \geq 0$.

Polynomial Functions

Factoring Polynomials

Factoring polynomials helps in finding zeros and simplifying expressions.

• Use techniques such as grouping, difference of squares, and trinomials.

Finding Real Zeros of Polynomials

Real zeros are the x-values where the polynomial equals zero.

• Set the polynomial equal to zero and solve for x, often by factoring.

Example: Find zeros of $x^2 - 5x + 6$:

• Factor: $(x - 2)(x - 3) = 0$

• Zeros: $x = 2, 3$

Rational Functions

Simplifying Rational Expressions

Reduce rational expressions to lowest terms by factoring and canceling common factors.

• Example: $\frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3} = x + 3$ (for $x \neq 3$)

Rationalizing the Denominator/Numerator

See "Rationalization Techniques" above for methods and examples.

Business Applications

Equilibrium Point

The equilibrium point is where the supply and demand functions are equal.

• Solve $\text{Supply} = \text{Demand}$ for $x$ and $p$.

Example: If $\text{Supply}: p = 2x + 5$ and $\text{Demand}: p = 20 - x$, set $2x + 5 = 20 - x$:

• $3x = 15 \implies x = 5$

• $p = 2 \times 5 + 5 = 15$

• Equilibrium point: $(5, 15)$

Appendix: Summary Table of Key Algebraic Properties

Property	Formula	Example
Product of Powers	$x^m x^n = x^{m+n}$	$2^3 \cdot 2^2 = 2^5 = 32$
Quotient of Powers	$\frac{x^n}{x^m} = x^{n-m}$	$\frac{5^4}{5^2} = 5^{2} = 25$
Power of a Power	$(x^n)^m = x^{nm}$	$(3^2)^3 = 3^{6} = 729$
Difference of Squares	$x^2 - a^2 = (x - a)(x + a)$	$x^2 - 16 = (x - 4)(x + 4)$
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$x^2 - 4x + 3 = 0$
Slope Formula	$m = \frac{y_2 - y_1}{x_2 - x_1}$	$(1,2), (3,8): m = 3$

Additional info: The study guide references specific textbook appendices and exercises for further practice, which are not included here but are recommended for mastery of these topics.