College Algebra Study Notes: Complex Numbers, Quadratic Equations, and Inequalities

Complex Numbers

Imaginary Unit and Complex Numbers

Complex numbers extend the real number system by introducing the imaginary unit i, defined as the square root of -1. This allows for the solution of equations that have no real solutions.

• Imaginary Unit: The number i is defined by $i = \sqrt{-1}$, so $i^2 = -1$.

• Complex Number: Any number of the form $a + bi$, where a and b are real numbers, is called a complex number. a is the real part, b is the imaginary part.

Examples

• Express $\sqrt{-25}$ in terms of i: $\sqrt{-25} = 5i$

• Express $\sqrt{-12}$: $\sqrt{-12} = 2\sqrt{3}i$

• Express $\sqrt{-13}$: $\sqrt{-13} = \sqrt{13}i$

Operations with Complex Numbers

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

• Multiplication: Use distributive property and $i^2 = -1$.

• Division: Multiply numerator and denominator by the conjugate of the denominator.

Examples

• Multiply $\sqrt{-9} \cdot \sqrt{-25} = (3i) \cdot (5i) = 15i^2 = -15$

• Add $(-2 - 4i) + (5 + 2i) - (3 - 6i) = (-2 + 5 - 3) + (-4i + 2i + 6i) = 0 + 4i$

• Multiply $(1 + 4i)(6 - 2i)$: Expand and simplify using $i^2 = -1$.

• Divide $\frac{2 - 5i}{1 - 6i}$: Multiply numerator and denominator by the conjugate $(1 + 6i)$.

Quadratic Equations

Definition and Forms

A quadratic equation is any equation that can be written in the form $ax^2 + bx + c = 0$, where $a \neq 0$. A quadratic function is $f(x) = ax^2 + bx + c$.

• Zero Product Property: If $AB = 0$, then $A = 0$ or $B = 0$.

Solving Quadratic Equations

• Factoring: Write the equation in factored form and use the zero product property.

• Square Root Property: If $x^2 = k$, then $x = \sqrt{k}$ or $x = -\sqrt{k}$.

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

• Quadratic Formula: For $ax^2 + bx + c = 0$, the solutions are: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Examples

• Solve $x^2 - 8x = 0$: Factor to $x(x - 8) = 0$, so $x = 0$ or $x = 8$.

• Solve $x^2 = 64$: $x = 8$ or $x = -8$.

• Solve $y^2 + 36 = 0$: $y^2 = -36$, so $y = 6i$ or $y = -6i$.

• Quadratic Formula example: Solve $6x^2 - 3x = 0$: $x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 6 \cdot 0}}{2 \cdot 6}$

Discriminant and Nature of Roots

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is $b^2 - 4ac$.

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

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

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

Example

• For $x^2 - 5x + 3 = 0$, discriminant is $(-5)^2 - 4 \cdot 1 \cdot 3 = 25 - 12 = 13 > 0$ (two real solutions).

More Equations and Applications

Solving Higher-Degree Equations

Equations of degree higher than two can often be solved by factoring, substitution, or using properties of exponents and radicals.

• Example: Solve $x^3 + 2x^2 - 12x - 9 = 0$ by factoring or synthetic division.

• Example: Solve $x^5 - 2x^2 = 16$ by isolating terms and factoring.

Applications: Word Problems

Algebraic equations are used to model real-world situations, such as motion problems or average calculations.

• Example: A boat travels downstream and upstream with different speeds due to current. Let $b$ be the speed in still water, $c$ the current speed. Use time = distance/speed to set up equations.

Absolute Value Equations and Inequalities

Properties and Solving Methods

Absolute value equations and inequalities involve expressions of the form $|E| = k$ or $|E| \leq k$, where $k$ is a nonnegative real number.

• Absolute Value Equation: $|x| = a$ if and only if $x = a$ or $x = -a$.

• Absolute Value Inequality:

  • $|x| \leq a$ if and only if $-a \leq x \leq a$

  • $|x| \geq a$ if and only if $x \leq -a$ or $x \geq a$

Examples

• Solve $|2t - 3| = 6$: $2t - 3 = 6$ or $2t - 3 = -6$

• Solve $|7x - 3| = 8$

• Solve $|2x - 5| = x + 1$

Linear, Compound, and Absolute Value Inequalities

Solving Linear and Compound Inequalities

Linear inequalities can be solved similarly to equations, but require careful attention to the direction of the inequality when multiplying or dividing by negative numbers. Compound inequalities involve two inequalities joined by 'and' or 'or'.

• Example: Solve $\frac{1}{2}x + 4 \leq \frac{3}{6}x - 2$

• Compound: $\frac{1}{2}x - 5 \leq 6$ or $x > 2$

Properties of Absolute Value Inequalities

• $|x| \leq a$ means $-a \leq x \leq a$

• $|x| \geq a$ means $x \leq -a$ or $x \geq a$

Examples

• Solve $|2x - 6| < 7$

• Solve $|4x - 2| \geq 3$

Applications: Average and Measurement Problems

Absolute value inequalities are used in practical contexts such as grading and measurement tolerances.

• Example: To achieve an average of at least 90, set up an inequality involving all test scores and solve for the unknown final exam score.

• Example: For a machine dispensing 8 oz with error no more than 0.05 oz, write $|x - 8| \leq 0.05$ and solve for $x$.

Summary Table: Discriminant and Nature of Roots

Discriminant ($b^2 - 4ac$)	Number/Nature of Solutions
> 0	Two real solutions
= 0	One real solution
< 0	Two nonreal (complex) solutions