College Algebra: Linear, Quadratic, Complex, and Radical Equations Study Guide

Section 1.1: Linear Equations

Definition and Properties of Linear Equations

A linear equation in one variable is an equation equivalent in form to $ax = b$, where $a$ and $b$ are real numbers and $x$ is the variable. The definition specifies $a \neq 0$ to ensure the equation is truly linear and not a constant equation.

• Solution Set: The set of all values of $x$ that make the equation true.

• Equivalent Equations: Equations with the same solution set.

• Solving Linear Equations: Use properties of equality, such as addition and multiplication, to isolate the variable.

Example: Solve $6 + y = 11$ by subtracting 6 from both sides to get $y = 5$.

Solving Linear Equations: Examples

• Simple Equations: $6 + y = 11$

• Fractional Equations: $\frac{1}{7}x = 4$

• Equations with Variables on Both Sides: $3x - 9 = -24$

• Equations with Distribution: $12 - 2x - 3(x + 2) = 4x + 12 - x$

• Decimal Equations: $0.9x - 1.2 = 0.4x + 0.1$

Equations That Lead to Linear Equations

Some equations may not appear linear initially but can be simplified to linear form.

• Example: $(x + 2)(x - 4) = (x + 3)^2$

• Example: $\frac{2x}{x+5} - \frac{x-6}{x+5} = 2$

Word Problems Modeled by Linear Equations

Many real-world problems can be modeled and solved using linear equations.

1. Determine what you are looking for.

2. Assign a variable to represent the unknown.

3. Translate the problem into a mathematical statement.

4. Solve the equation and answer the question in a complete sentence.

5. Check your answer with the facts presented.

Example: If $15,000$ is to be divided between Jon and Wendy, with Jon to receive $2,500$ less than Wendy, set up and solve a linear equation to find their shares.

Section 1.2: Quadratic Equations

Definition and Standard Form

A quadratic equation is an equation equivalent to $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$. The standard form is $ax^2 + bx + c = 0$.

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

• Discriminant: $b^2 - 4ac$ determines the nature of the roots.

Methods for Solving Quadratic Equations

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

• Square Root Method: Used when the equation is in the form $x^2 = p$.

• Completing the Square: Transform the equation into $(x + d)^2 = p$ and solve for $x$.

• Quadratic Formula: Use when factoring is not possible or convenient.

Examples of Solving Quadratic Equations

• Factoring: $x^2 + 2x - 15 = 0$

• Square Root Method: $x^2 - 7 = 0$

• Completing the Square: $x^2 + 6x + 5 = 0$

• Quadratic Formula: $2x^2 - 4x + 1 = 0$

Discriminant and Nature of Roots

• If $b^2 - 4ac > 0$, two distinct real solutions.

• If $b^2 - 4ac = 0$, one real repeated solution.

• If $b^2 - 4ac < 0$, two complex solutions.

Word Problems Modeled by Quadratic Equations

Some applied problems, such as finding the dimensions of a box with a given volume, require setting up and solving a quadratic equation.

Section 1.3: Complex Numbers; Quadratic Equations in the Complex Number System

Definition and Properties of Complex Numbers

A complex number is of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, defined by $i^2 = -1$.

• Imaginary Unit: $i = \sqrt{-1}$

• Standard Form: $a + bi$

• Equality: $a + bi = c + di$ if and only if $a = c$ and $b = d$

Operations with Complex Numbers

• Addition/Subtraction: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

• Conjugate: The conjugate of $a + bi$ is $a - bi$

• Division: Multiply numerator and denominator by the conjugate of the denominator to express in standard form.

Powers of i

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

• Powers of $i$ repeat every four terms.

Quadratic Equations in the Complex Number System

When $b^2 - 4ac < 0$, quadratic equations have complex solutions. The quadratic formula can be used in the complex number system without restriction.

• Principal Square Root: For $N > 0$, $\sqrt{-N} = i\sqrt{N}$

• Character of Solutions:

  • Two unequal real solutions if $b^2 - 4ac > 0$

  • One repeated real solution if $b^2 - 4ac = 0$

  • Two complex conjugate solutions if $b^2 - 4ac < 0$

Section 1.4: Radical Equations; Equations Quadratic in Form; Factorable Equations

Solving Radical Equations

A radical equation is an equation in which the variable occurs under a radical sign (such as a square root).

• Step 1: Isolate the radical.

• Step 2: Raise both sides to the power of the index (e.g., square both sides for a square root).

• Step 3: Solve the resulting equation.

• Step 4: Check all solutions in the original equation to avoid extraneous solutions.

Example: Solve $\sqrt{x} - 3 = 1$ by isolating $\sqrt{x}$, squaring both sides, and solving for $x$.

Equations Quadratic in Form

Some equations can be transformed into quadratic equations by substitution.

• Example: $x^4 + 4x^2 - 9 = 0$ can be solved by letting $u = x^2$ and solving $u^2 + 4u - 9 = 0$.

Factoring Equations

Factoring is a method used to solve equations by expressing them as a product of factors and setting each factor to zero.

• Example: $x^4 + 4x^2 - 9 = 36$

Summary Table: Methods for Solving Equations

Equation Type	Standard Form	Solution Methods
Linear	$ax = b$	Isolate variable, properties of equality
Quadratic	$ax^2 + bx + c = 0$	Factoring, square root, completing the square, quadratic formula
Complex	$a + bi$	Add/subtract, multiply, divide, conjugate
Radical	Variable under radical	Isolate radical, raise to power, solve, check

Additional info: Some steps and definitions were expanded for clarity and completeness, including the summary table and detailed solution steps for each equation type.