College Algebra: Essential Concepts and Problem-Solving Strategies

Linear Equations and Their Solutions

Definition of a Linear Equation in One Variable

A linear equation in one variable is an equation that can be written in the form

$ax + b = 0$

• a and b are real numbers, and a is not zero.

Solving a Linear Equation

1. Simplify each side by removing grouping symbols and combining like terms.

2. Collect all variable terms on one side and constants on the other.

3. Isolate the variable and solve.

4. Check the solution in the original equation.

The Least Common Denominator (LCD)

The least common denominator is the smallest polynomial that is a multiple of all denominators in a rational expression, found by taking the product of all prime factors with the highest exponent present.

Nature of the Solutions of a Linear Equation

• Conditional equation: $x =$ a real number (one solution)

• Identity: $x =$ all real numbers (infinitely many solutions)

• Inconsistent equation: $x =$ no real numbers (no solution)

Strategy for Solving Word Problems

1. Read the problem carefully and identify what is given and what is to be found.

2. Write expressions for unknowns in terms of a variable.

3. Write an equation modeling the conditions.

4. Solve the equation and answer the question.

5. Check the solution in the context of the problem.

Geometric Formulas

Common Area, Perimeter, and Volume Formulas

• Square: $A = s^2$, $P = 4s$

• Rectangle: $A = lw$, $P = 2l + 2w$

• Circle: $A = \pi r^2$, $C = 2\pi r$

• Triangle: $A = \frac{1}{2}bh$

• Trapezoid: $A = \frac{1}{2}(a + b)h$

• Cube: $V = s^3$

• Rectangular Solid: $V = lwh$

• Circular Cylinder: $V = \pi r^2 h$

• Sphere: $V = \frac{4}{3}\pi r^3$

• Cone: $V = \frac{1}{3}\pi r^2 h$

Complex Numbers and Imaginary Numbers

The Imaginary Unit

• The imaginary unit $i$ is defined as $\sqrt{-1} = i$, so $i^2 = -1$.

Complex Numbers

• The set of all numbers in the form $a + bi$, where $a$ and $b$ are real numbers, is called the set of complex numbers.

• Real part: $a$

• Imaginary part: $b$

• If $b = 0$, the number is real; if $a = 0$ and $b \neq 0$, the number is a pure imaginary number.

Adding and Subtracting Complex Numbers

• $(a + bi) + (c + di) = (a + c) + (b + d)i$

• $(a + bi) - (c + di) = (a - c) + (b - d)i$

Multiplying Complex Numbers (FOIL Method)

• Use the distributive property or FOIL method: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Conjugate of a Complex Number

• The conjugate of $a + bi$ is $a - bi$.

• $(a + bi)(a - bi) = a^2 + b^2$ (always a real number).

Division of Complex Numbers

• To divide, multiply numerator and denominator by the conjugate of the denominator.

Principal Square Root of a Negative Number

• For $-b$ ($b > 0$), $\sqrt{-b} = i\sqrt{b}$

Quadratic Equations

Definition

• A quadratic equation in $x$ is an equation that can be written in the form $ax^2 + bx + c = 0$, where $a \neq 0$.

The Zero-Product Principle

• If $AB = 0$, then $A = 0$ or $B = 0$.

Solving Quadratic Equations by Factoring

1. Rewrite in standard form $ax^2 + bx + c = 0$.

2. Factor completely.

3. Set each factor to zero and solve.

4. Check solutions in the original equation.

Solving Quadratic Equations by the Square Root Property

• If $x^2 = d$, then $x = \pm \sqrt{d}$.

• If $x^2 = -d$, then $x = \pm i\sqrt{d}$.

Perfect Square Trinomials

• $(A + B)^2 = A^2 + 2AB + B^2$

• $(A - B)^2 = A^2 - 2AB + B^2$

Difference of Two Squares

• $(A + B)(A - B) = A^2 - B^2$

Completing the Square

1. If $ax^2 + bx + c = 0$, divide both sides by $a$ if $a \neq 1$.

2. Move the constant to the other side.

3. Add $\left(\frac{b}{2a}\right)^2$ to both sides.

4. Factor and solve for $x$.

The Quadratic Formula

• The solutions to $ax^2 + bx + c = 0$ are given by:

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

The Discriminant

• The discriminant is $b^2 - 4ac$.

• It determines the number and type of solutions:

Discriminant $b^2 - 4ac$	Kinds of Solutions	Graph of $ax^2 + bx + c = 0$
> 0	Two unequal real solutions	Parabola crosses x-axis twice
= 0	One real solution (repeated root)	Parabola touches x-axis once
< 0	No real solution; two complex solutions	Parabola does not cross x-axis

Radical Equations and Rational Exponents

Solving Radical Equations Containing $n$th Roots

1. Isolate the radical.

2. Raise both sides to the $n$th power.

3. Solve the resulting equation.

4. Check all solutions in the original equation.

Definition of a Rational Exponent

• $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Solving Equations of the Form $x^{m/n} = k$

1. Isolate $x$ with the rational exponent.

2. Raise both sides to the $n/m$ power.

3. Check all solutions in the original equation.

Quadratic Equations in Quadratic Form

Definition

• An equation is in quadratic form if it can be written as $au^2 + bu + c = 0$ for some expression $u$.

Rewriting Absolute Value Equations

• If $c$ is positive and $|u| = c$, then $u = c$ or $u = -c$.

Set Notation and Intervals

Set-Builder Notation

• Describes a set by stating the property its members must satisfy, e.g., $\{x | 0 \leq x \leq 30\}$.

Interval Notation

• Parentheses $( )$ exclude endpoints; brackets $[ ]$ include endpoints.

Interval	Set-Builder	Graph
$[a, b]$	$a \leq x \leq b$	Solid dots at $a$ and $b$
$(a, b)$	$a < x < b$	Open circles at $a$ and $b$
$[a, b)$	$a \leq x < b$	Solid dot at $a$, open at $b$
$(a, b]$	$a < x \leq b$	Open at $a$, solid at $b$
$(-\infty, a)$	$x < a$	Open at $a$
$(a, \infty)$	$x > a$	Open at $a$

Finding Intersections and Unions of Intervals

1. Graph intervals on a number line.

2. Intersection: portion common to both intervals.

3. Union: total collection covered by either interval.

Inequalities

Properties of Inequality

Property	The Property in Words	Example
Addition	If the same amount is added/subtracted to both sides, the inequality remains equivalent.	Subtract 2: $2x + 3 < 7 \rightarrow 2x < 4$
Positive Multiplication	If both sides are multiplied/divided by a positive number, the inequality remains equivalent.	Divide by 2: $2x < 4 \rightarrow x < 2$
Negative Multiplication	If both sides are multiplied/divided by a negative number, the inequality reverses direction.	Divide by -4: $-4x > 20 \rightarrow x < -5$

Solving Absolute Value Inequalities

• If $|x| < c$, then $-c < x < c$

• If $|x| > c$, then $x < -c$ or $x > c$

Pythagorean Theorem

Right Triangle Relationship

• For a right triangle with legs $a$ and $b$, and hypotenuse $c$:

$a^2 + b^2 = c^2$

Summary

• Linear and quadratic equations, complex numbers, inequalities, and interval notation are foundational topics in College Algebra.

• Mastery of these concepts is essential for solving a wide range of algebraic problems.