Comprehensive Study Guide: College Algebra Concepts and Problem Solving

Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is raised only to the power of one. Solving these equations involves isolating the variable on one side of the equation.

• Definition: An equation of the form ax + b = c, where a, b, and c are constants.

• Steps:

  1. Combine like terms on each side.

  2. Isolate the variable using inverse operations (addition, subtraction, multiplication, division).

  3. Check the solution by substituting back into the original equation.

• Example: Solve $4n + 6 = 10$.

  • Subtract 6: $4n = 4$

  • Divide by 4: $n = 1$

Solving Inequalities

Inequalities express a relationship where two expressions are not necessarily equal, using symbols such as <, >, ≤, or ≥. The solution is often a range of values.

• Definition: An inequality is a mathematical statement that relates expressions that are not equal.

• Steps:

  1. Solve similarly to equations, but reverse the inequality sign when multiplying or dividing by a negative number.

  2. Express the solution as an interval or set.

• Example: Solve $3x + 1 > 5$.

  • Subtract 1: $3x > 4$

  • Divide by 3: $x > \frac{4}{3}$

Complex Numbers

Operations with Complex Numbers

Complex numbers are numbers in the form a + bi, where i is the imaginary unit ($i^2 = -1$).

• Addition/Subtraction: Combine real parts and imaginary parts separately.

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

• Example (Addition): $(9 + 6i) + (6 + 7i) = 15 + 13i$

• Example (Multiplication): $(9i)(8 - 2i) = 72i - 18i^2 = 72i + 18 = 18 + 72i$

Exponents and Radicals

Properties of Exponents

Exponents represent repeated multiplication. Several rules govern their manipulation.

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Zero Exponent: $a^0 = 1$ (for $a \neq 0$)

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

• Example: $\frac{7^3}{7^5} = 7^{-2} = \frac{1}{49}$

Radicals and Rational Exponents

Radicals can be expressed as rational exponents. The $n$th root of $a$ is $a^{1/n}$.

• Example: $\sqrt[3]{8} = 8^{1/3} = 2$

• Example: $16^{3/4} = (16^{1/4})^3 = (2)^3 = 8$

Quadratic Equations

Solving by Square Root Property

The square root property is used when the equation is in the form $x^2 = k$.

• Formula: If $x^2 = k$, then $x = \pm \sqrt{k}$

• Example: $x^2 = 25 \implies x = \pm 5$

Completing the Square

This method rewrites a quadratic equation in the form $(x + p)^2 = q$ to solve for $x$.

• Steps:

  1. Move the constant to the other side.

  2. Add $(\frac{b}{2})^2$ to both sides to complete the square.

  3. Take the square root of both sides and solve for $x$.

• Example: $x^2 + 4x + 9 = 0$

  • Move 9: $x^2 + 4x = -9$

  • Add 4: $x^2 + 4x + 4 = -9 + 4$

  • $(x + 2)^2 = -5$

  • $x + 2 = \pm i\sqrt{5}$

  • $x = -2 \pm i\sqrt{5}$

Quadratic Formula

The quadratic formula solves any quadratic equation of the form $ax^2 + bx + c = 0$.

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

• Discriminant: $b^2 - 4ac$ determines the nature of the roots (real and distinct, real and equal, or complex).

• Example: $2x^2 + 12x + 18 = 0$

  • $a = 2, b = 12, c = 18$

  • $x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 2 \cdot 18}}{2 \cdot 2}$

Functions and Applications

Modeling with Functions

Functions can model real-world situations, such as population growth. The solution often involves substituting values and solving for the variable.

• Example: Given $P = 0.67x^2 - 0.084x + 2$ (population in thousands), find $x$ when $P = 25.85$.

Distance and Geometry in the Coordinate Plane

Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

• Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Example: Between $(2, 6)$ and $(5, -6)$:

  • $d = \sqrt{(5-2)^2 + ((-6)-6)^2} = \sqrt{9 + 144} = \sqrt{153} \approx 12.37$

Pythagorean Theorem

Used to determine if three points form a right triangle by checking if the sum of the squares of two sides equals the square of the third side.

• Formula: $a^2 + b^2 = c^2$

Circles in the Coordinate Plane

Equation of a Circle

The standard form of the equation of a circle with center $(h, k)$ and radius $r$ is:

• Formula: $(x - h)^2 + (y - k)^2 = r^2$

• Example: Center at $(0, 0)$, radius $6$:$x^2 + y^2 = 36$

Finding Center and Radius

Given an equation, rewrite it in standard form to identify the center and radius.

• Example: $x^2 + y^2 - 8x + 6y = 0$

  • Complete the square for $x$ and $y$ terms.

Graphing Circles

Plot the center and use the radius to draw the circle. The equation can be derived from the graph by identifying the center and radius.

Quadrilaterals and Rectangles in the Plane

Determining Rectangles

Given four points, determine if they form a rectangle by checking if opposite sides are equal and adjacent sides are perpendicular (using slopes).

• Slopes: Perpendicular lines have slopes that are negative reciprocals.

• Distances: Opposite sides must be equal in length.

Summary Table: Key Formulas and Properties

Concept	Formula/Property	Example
Linear Equation	$ax + b = c$	$2x + 3 = 7$
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$x^2 - 4x + 3 = 0$
Distance Formula	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	$(1,2)$ and $(4,6)$
Circle Equation	$(x - h)^2 + (y - k)^2 = r^2$	Center $(2,3)$, $r=5$
Exponent Rules	$a^m \cdot a^n = a^{m+n}$	$2^3 \cdot 2^2 = 2^5$