Precalculus Study Notes: Complex Numbers, Quadratic Equations, and 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: $3 + 4i$ is a complex number with real part 3 and imaginary part 4.

Operations with Complex Numbers

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

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

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

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

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

$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

• Example: $\frac{4i}{3 - 9i}$

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 Standard Form

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

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

• Example: $2x^2 - 3x + 1 = 0$

Solving Quadratic Equations

• Factoring: Express the quadratic as a product of two binomials and use the zero-product principle.

• 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: For $ax^2 + bx + c = 0$:

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

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

The Discriminant and Types of Solutions

The discriminant $b^2 - 4ac$ indicates the number and type of solutions for a quadratic equation:

Discriminant $b^2 - 4ac$	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 (vertex touches x-axis)
$b^2 - 4ac < 0$	No real solution; two complex conjugate solutions	No x-intercepts

Section 1.6: Other Types of Equations

Polynomial, Radical, and Rational Exponent Equations

Equations may involve higher-degree polynomials, radicals, or rational exponents. These can often be solved by factoring, isolating the variable, or making substitutions to reduce them to quadratic form.

• Polynomial Equations: Set equal to zero and factor.

• Radical Equations: Isolate the radical and square both sides, checking for extraneous solutions.

• Rational Exponents: Rewrite using radicals and solve as above.

Section 1.7: Linear Inequalities and Absolute Value Equations

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: $2x - 3 < 7$ leads to $x < 5$

Absolute Value Equations and Inequalities

Absolute value equations are rewritten as two separate equations:

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

• If $|u| < c$, then $-c < u < c$

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

Additional Topic: The Pythagorean Theorem

Right Triangles and the Pythagorean Theorem

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:

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

• Application: Used to find the length of a side in a right triangle when the other two sides are known.