College Algebra Unit Test 1 Review: Key Concepts and Study Guide

Complex Numbers

Definition and Properties

Complex numbers extend the real number system by including the imaginary unit i, where $i^2 = -1$. They are written in the form $a + bi$, where a and b are real numbers.

• Imaginary Unit: $i = \sqrt{-1}$

• Pure Imaginary Numbers: Numbers of the form $bi$ where $b \neq 0$

• Complex Conjugate: For $z = a + bi$, the conjugate is $\overline{z} = a - bi$

• Multiplication and Division: Use the distributive property and rationalize denominators as needed.

Example: $(2 + 3i) + (4 - 5i) = 6 - 2i$

Quadratic Equations

Solving Quadratics

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula.

• Factoring: Express as a product of binomials and set each factor to zero.

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

• Zero Factor Property: If $ab = 0$, then $a = 0$ or $b = 0$

• Discriminant: $b^2 - 4ac$ determines the nature of the roots (real and distinct, real and equal, or complex).

Example: Solve $x^2 - 5x + 6 = 0$ by factoring: $(x-2)(x-3) = 0 \implies x = 2, 3$

Common Applications of Quadratic Equations

Word Problems and Modeling

Quadratic equations are used to model various real-world scenarios, such as projectile motion, area problems, and optimization.

• Setting up Equations: Translate word problems into quadratic equations.

• Solving for Unknowns: Use appropriate methods to find solutions relevant to the context.

Example: The area of a rectangle is 45 square inches. If the long side is 1 inch longer than the short side, set up and solve $x(x+1) = 45$.

Linear Equations and Inequalities

Solving and Graphing

Linear equations are of the form $ax + b = 0$. Linear inequalities involve expressions like $ax + b > 0$ or $ax + b < 0$.

• Solving Linear Equations: Isolate the variable using inverse operations.

• Solving Linear Inequalities: Similar to equations, but reverse the inequality when multiplying or dividing by a negative number.

• Graphing Solutions: Represent solutions on a number line or coordinate plane.

• Intersection of Intervals: Find where two or more inequalities are simultaneously true.

Example: Solve $2x - 3 > 5 \implies x > 4$

Basics of Functions & Their Graphs

Definitions and Properties

A function is a relation that assigns exactly one output to each input. Key features include domain, range, and the vertical line test.

• Domain: Set of all possible input values (x-values).

• Range: Set of all possible output values (y-values).

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

• Evaluating Functions: Substitute the input value into the function rule.

• Finding Information from Graphs: Use the graph to determine function values, intercepts, and intervals of increase/decrease.

Example: For $f(x) = x^2$, the domain is all real numbers, and the range is $[0, \infty)$.

Challenging Problems

Sample Problems

• Evaluate $\frac{4 + 3i}{2 - i}$ and simplify your answer.

• Given a rectangle with area 45 square inches and the long side 1 inch longer than the short side, find the dimensions.

• Solve $(x^2 - 6x^2 - 1)(|x| - 2) + 2 = 0$.

Summary Table: Key Concepts

Topic	Main Concepts	Key Formula/Property
Complex Numbers	Imaginary unit, arithmetic, conjugates	$i^2 = -1$
Quadratic Equations	Factoring, quadratic formula, discriminant	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Linear Equations/Inequalities	Solving, graphing, intersection of intervals	Isolate variable, reverse inequality for negatives
Functions & Graphs	Domain, range, vertical line test	Each input has one output

Study Tips

• Start with topics you find most challenging.

• Practice regularly and review mistakes.

• Work with classmates and use available resources.

• Attempt challenging problems for deeper understanding.