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 $\overline{z} = a - bi$

• Operations: Addition, subtraction, multiplication, and division follow algebraic rules, using $i^2 = -1$ to simplify.

Example: $(2 + 3i) + (1 - 4i) = 3 - i$

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 $(x - r_1)(x - r_2) = 0$ and solve for $x$.

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

• Completing the Square: Rewrite $ax^2 + bx + c = 0$ in the form $(x + d)^2 = e$ and solve for $x$.

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

Example: Solve $x^2 - 4x + 3 = 0$ by factoring: $(x-1)(x-3) = 0 \implies x = 1, 3$

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.

• Area Problems: Setting up equations based on geometric relationships.

• Projectile Motion: Height as a function of time: $h(t) = -16t^2 + vt + h_0$

• Maxima/Minima: The vertex of the parabola $x = -\frac{b}{2a}$ gives the maximum or minimum value.

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 use $<$, $>$, $\leq$, or $\geq$ instead of $=$.

• Solving Linear Equations: Isolate $x$ using inverse operations.

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

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

• Interval Notation: Use $(a, b)$ for open intervals, $[a, b]$ for closed intervals.

Example: Solve $2x - 5 < 7 \implies x < 6$

Basics of Functions & Their Graphs

Definitions and Properties

A function is a relation that assigns exactly one output to each input. The domain is the set of possible inputs, and the range is the set of possible outputs.

• Function Notation: $f(x)$ denotes the output for input $x$.

• 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 Domain and Range: Identify all valid $x$ (domain) and $f(x)$ (range) values.

Example: For $f(x) = x^2 + 1$, the domain is all real numbers, and the range is $[1, \infty)$.

Challenging Problems

Sample Problems

• Evaluate: $\frac{4 + 3i}{2 - i}$ (Simplify your answer)

• Rectangle Problem: If the long side is 1 inch longer than the short side and the area is 45 square inches, find the dimensions.

• Solve: $(x^2 - 2x) - [(1 - 2i)^2 + 2i] = 0$

Study Tips

• Start with topics you find most challenging.

• Use available resources: textbooks, online tools, study groups, and office hours.

• Practice regularly and review mistakes to reinforce understanding.

• Work with classmates to discuss and solve different types of problems.

• Simulate test conditions with practice exams to build confidence.

Suggested Exercises

Chapter	Section	Exercises
1	Section 4	9-12, 17-21, 25-27, 31-35
1	Section 5	1-8, 13-16, 21-26, 31-36, 45-47, 51-56, 57-64, 101, 141, 145, 146, 147, 148
1	Section 6	1-8, 17-24, 47-56, 77-84, 119
2	Section 1	1-16, 19-32, 37-56, 57-62, 67-70