Quadratic Functions, Equations, and Inequalities: Study Notes for College Algebra (Precalculus)

Complex Numbers

The Imaginary Number

The imaginary unit, denoted as i, is defined by the property:

• i is the number such that $i^2 = -1$.

• This definition allows us to extend the square root operation to negative numbers: $\sqrt{-a} = i\sqrt{a}$ for $a > 0$.

Example: Simplify $\sqrt{-36} = 6i$.

Example: Simplify $\sqrt{-16} = 4i$.

Complex Numbers

A complex number is any number of the form $a + bi$, where $a$ and $b$ are real numbers. Here, $a$ is the real part and $b$ is the imaginary part.

• Example: $5 - 4i$ has real part 5 and imaginary part -4.

Addition and Subtraction of Complex Numbers

To add or subtract complex numbers, combine like terms (real with real, imaginary with imaginary):

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

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

• Example: $(-5 + 6i) - (6 + 9i) = (-11 - 3i)$

Multiplication of Complex Numbers

Use the distributive law and replace $i^2$ with $-1$:

• Example: $(1 + i\sqrt{3})(\sqrt{3} + i)$

• Multiply out and simplify, remembering $i^2 = -1$.

Complex Conjugate and Division

The conjugate of $a + bi$ is $a - bi$. The product of a complex number and its conjugate is always real:

• $(a + bi)(a - bi) = a^2 + b^2$

• Example: $2 - 3i$ and $2 + 3i$ are conjugates.

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

• Example: $\frac{5 - 3i}{3 - 4i}$

Powers of i

Powers of $i$ repeat every four:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

• For $n \geq 4$, $i^n = i^r$ where $r$ is the remainder when $n$ is divided by 4.

Quadratic Equations, Functions, and Models

Definition of Quadratic Functions and Equations

A quadratic function is $f(x) = ax^2 + bx + c$, $a \neq 0$. A quadratic equation is $ax^2 + bx + c = 0$, $a \neq 0$.

• The zeros of $f(x)$ are the solutions to $ax^2 + bx + c = 0$.

Equation-Solving Principles

• Zero Product Principle: If $ab = 0$, then $a = 0$ or $b = 0$.

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

Solving Quadratic Equations by Factoring

1. Set the equation to zero.

2. Factor completely.

3. Set each factor to zero and solve.

Example: $(x + 2)(2x - 1) = 0$ yields $x = -2$ or $x = \frac{1}{2}$.

Solving Quadratic Equations by Square Roots

Isolate the squared term and apply the square root principle.

• Example: $2(x + 3)^2 + 1 = 9$

Completing the Square

To solve $x^2 + bx = x + \frac{b^2}{4} - \frac{b^2}{4}$, add $\frac{b^2}{4}$ to both sides to form a perfect square:

• $x^2 + bx + \frac{b^2}{4} = (x + \frac{b}{2})^2$

Example: Solve $2x^2 - 4x + 1 = 0$ by completing the square.

The Quadratic Formula

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

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

• Example: Solve $3x^2 + 2x - 7 = 0$ using the quadratic formula.

The Discriminant

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

• If $b^2 - 4ac < 0$: Two different imaginary solutions.

• If $b^2 - 4ac = 0$: One real solution (repeated root).

• If $b^2 - 4ac > 0$: Two different real solutions.

Equations Reducible to Quadratic Form

Some equations can be solved by substitution to make them quadratic.

• Example: $x^4 - 5x^2 + 4 = 0$ (let $u = x^2$)

Applications of Quadratic Equations

• Word problems involving area, motion, and geometry can often be modeled with quadratic equations.

• Example: Two trains leave a station at the same time, one west and one south, and are 200 km apart after 2 hours. Find their speeds.

Analyzing the Graphs of Quadratic Functions

Vertex Form of a Quadratic Function

Any quadratic function $f(x) = ax^2 + bx + c$ can be rewritten as $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

• Completing the square is used to convert to vertex form.

Graphing Quadratic Functions

The graph of $f(x) = a(x - h)^2 + k$ is a parabola:

• Horizontal translation: $h$ units right (if $h > 0$) or left (if $h < 0$)

• Vertical translation: $k$ units up (if $k > 0$) or down (if $k < 0$)

• Vertical stretch/shrink: $|a|$ > 1 stretches, $0 < |a| < 1$ shrinks

• Reflection: If $a < 0$, the parabola opens downward

The Vertex Formula

For $f(x) = ax^2 + bx + c$:

• Vertex: $h = \frac{-b}{2a}$, $k = f(h) = c - \frac{b^2}{4a}$

• Axis of symmetry: $x = h$

• If $a > 0$, the vertex is a minimum; if $a < 0$, the vertex is a maximum.

Solving Rational Equations

To solve rational equations:

1. Factor denominators and find the least common denominator (LCD).

2. Multiply both sides by the LCD to clear denominators.

3. Solve the resulting equation.

4. Check for extraneous solutions (values that make any denominator zero).

• Example: $\frac{x}{x+1} = \frac{3}{2(x-1)}$

Solving Radical Equations

A radical equation contains variables inside a radical. To solve:

1. Isolate the radical.

2. Raise both sides to the power matching the index of the radical.

3. Solve the resulting equation.

4. Check for extraneous solutions.

• Example: $\sqrt{7 + x} - 5 = x$

Equations with Absolute Values

The absolute value of $x$, $|x|$, is the distance from $x$ to zero on the number line:

• $|x| = x$ if $x \geq 0$

• $|x| = -x$ if $x < 0$

To solve $|A| = b$ ($b \geq 0$), set $A = b$ or $A = -b$.

• If $b < 0$, there is no solution.

• Example: $|6x| = 18$ yields $x = 3$ or $x = -3$.

For inequalities:

• $|A| \leq b$ becomes $-b \leq A \leq b$ ("and" compound inequality)

• $|A| \geq b$ becomes $A \leq -b$ or $A \geq b$ ("or" compound inequality)

References

• Bittinger, Algebra & Trigonometry, Graphs and Models, 6th Edition, Pearson.