Complex Numbers, Quadratic Equations, and Linear Inequalities

Section 1.4: Complex Numbers

The 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. Every complex number can be written in the form a + bi, where a is the real part and b is the imaginary part.

• Imaginary Unit: $i = \sqrt{-1}$

• Standard Form: $a + bi$

• Example: $-5 + i$ and $-11 - 6i$ are both complex numbers.

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided using algebraic rules, with the additional property that $i^2 = -1$.

• Addition/Subtraction: Combine like terms (real with real, imaginary with imaginary).

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

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

• Example: $(5 + 3i)(5 - 3i) = 25 - 9i^2 = 25 + 9 = 34$

Principal Square Root of a Negative Number

The principal square root of a negative number is defined using the imaginary unit:

• $\sqrt{-b} = i\sqrt{b}$, where $b > 0$

• Example: $\sqrt{-16} = 4i$

Section 1.5: Quadratic Equations

Definition and Forms

A quadratic equation is a second-degree polynomial equation in the form:

• $ax^2 + bx + c = 0$, where $a \neq 0$

• If $a = 0$, the equation becomes linear.

Zero-Product Principle

If the product of two expressions is zero, at least one of the expressions must be zero:

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

Solving Quadratic Equations

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

• Square Root Property: If $u^2 = d$, then $u = \pm\sqrt{d}$

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

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

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

The Discriminant and Types of Solutions

The discriminant $b^2 - 4ac$ determines the nature of the solutions:

Discriminant	Kinds of Solutions	Graph
$b^2 - 4ac > 0$	Two unequal real solutions (rational or irrational)	Two x-intercepts
$b^2 - 4ac = 0$	One real solution (repeated root)	One x-intercept
$b^2 - 4ac < 0$	Two imaginary solutions (complex conjugates)	No x-intercepts

Section 1.6: Other Types of Equations

Polynomial Equations

Polynomial equations can often be solved by factoring and applying the zero-product principle.

• Example: $9x^3 + 8 = 4x + 18x^2$

Radical Equations

Equations involving roots can be solved by isolating the radical and then raising both sides to the appropriate power. Always check for extraneous solutions.

• Example: $\sqrt{x + 1} = x - 1$

Equations with Rational Exponents

Rewrite rational exponents as roots and solve as radical equations.

• Example: $(x + 2)^{2/3} = 27$

Section 1.7: Linear Inequalities and Absolute Value Equations

Solving Absolute Value Equations

Absolute value equations can be rewritten as two separate equations:

• If $|u| = c$, then $u = c$ or $u = -c$

Solving Linear Inequalities

Linear inequalities are solved similarly to equations, but the direction of the inequality reverses when multiplying or dividing by a negative number. Solutions are often expressed in interval notation.

• Example: $2x - 3 < 11$

Solving Absolute Value Inequalities

• If $|u| < c$, then $-c < u < c$

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

Additional Topic: The Pythagorean Theorem

Right Triangle Relationships

The Pythagorean Theorem relates the lengths of the sides of a right triangle. If the legs have lengths a and b, and the hypotenuse has length c, then:

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