College Algebra Study Guide: Linear Equations, Quadratic Equations, and Complex Numbers

Linear Equations

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power higher than one. Solving these equations involves isolating the variable on one side of the equation.

• Definition: A linear equation is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants.

• Steps to Solve:

  1. Simplify both sides of the equation (expand, combine like terms).

  2. Isolate the variable by adding, subtracting, multiplying, or dividing both sides as needed.

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

• Example: Solve $13x - 9 = 6(2x + 3) - 9$

  • Expand: $13x - 9 = 12x + 18 - 9$

  • Simplify: $13x - 9 = 12x + 9$

  • Subtract $12x$ from both sides: $x - 9 = 9$

  • Add $9$ to both sides: $x = 18$

Quadratic Equations

Solving Quadratic Equations

Quadratic equations are equations where the variable is raised to the second power. These equations can be solved by factoring, completing the square, or using the quadratic formula.

• Definition: A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$.

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

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

• Example: Solve $x^2 - 17x + 72 = 0$

  • Factor: $(x - 8)(x - 9) = 0$

  • Solutions: $x = 8$, $x = 9$

Properties of Exponents

Simplifying Expressions with Exponents

Exponents indicate repeated multiplication. Understanding their properties is essential for simplifying algebraic expressions.

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

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

• Power Rule: $(a^m)^n = a^{mn}$

• Example: Simplify $3x^7y^7 + 12x^7y^7 - 30x^6y^6$

  • Combine like terms: $3x^7y^7 + 12x^7y^7 = 15x^7y^7$

  • Final expression: $15x^7y^7 - 30x^6y^6$

Complex Numbers

Definition and Operations

Complex numbers extend the real numbers by including the imaginary unit $i$, where $i^2 = -1$. They are written in the form $a + bi$, where $a$ and $b$ are real numbers.

• Definition: A complex number is any number of the form $a + bi$.

• Imaginary Unit: $i$ is defined such that $i^2 = -1$.

• Powers of $i$:

  • $i^1 = i$

  • $i^2 = -1$

  • $i^3 = -i$

  • $i^4 = 1$

  • Pattern repeats every four powers.

• Example: Calculate $i^5$

  • $i^5 = i^{4} \cdot i = 1 \cdot i = i$

Simplifying Complex Expressions

Complex expressions can be simplified using the properties of $i$ and by combining like terms.

• Example: Simplify $\sqrt{-3} \cdot \sqrt{-75}$

  • $\sqrt{-3} = i\sqrt{3}$, $\sqrt{-75} = i\sqrt{75}$

  • Product: $i\sqrt{3} \cdot i\sqrt{75} = i^2 \sqrt{3 \cdot 75} = -1 \cdot \sqrt{225} = -15$

• Example: Simplify $15i\sqrt{27}$

  • $\sqrt{27} = 3\sqrt{3}$

  • $15i\sqrt{27} = 15i \cdot 3\sqrt{3} = 45i\sqrt{3}$

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 real parts and imaginary parts separately.

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

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

• Example: $(4 - 3i)^2$

  • Expand: $(4 - 3i)^2 = 16 - 24i + 9i^2 = 16 - 24i + 9(-1) = 16 - 24i - 9 = 7 - 24i$

• Example: $\frac{3 + 5i}{5 + 6i}$

  • Multiply numerator and denominator by conjugate $(5 - 6i)$:

  • Numerator: $(3 + 5i)(5 - 6i) = 15 - 18i + 25i - 30i^2 = 15 + 7i + 30 = 45 + 7i$

  • Denominator: $(5 + 6i)(5 - 6i) = 25 - 36i^2 = 25 + 36 = 61$

  • Result: $\frac{45 + 7i}{61}$

Quadratic Equations with Complex Solutions

Solving Quadratics with No Real Solutions

Some quadratic equations have solutions that are not real numbers. In such cases, the solutions are complex numbers.

• Example: Solve $x^2 = 25$

  • Solutions: $x = 5$, $x = -5$

• Example: Solve $x^2 = -25$

  • Solutions: $x = 5i$, $x = -5i$

Summary Table: Powers of i

Power	Value
$i^1$	$i$
$i^2$	$-1$
$i^3$	$-i$
$i^4$	$1$
$i^5$	$i$
$i^6$	$-1$
$i^7$	$-i$
$i^8$	$1$

Additional info:

• Some problems involve multiple choice questions on powers of $i$ and simplification of complex numbers, which are standard topics in College Algebra.

• Problems also include solving quadratic equations by factoring and using the quadratic formula, as well as operations with exponents and polynomials.