Solving Linear Equations

Definition and Properties

Linear equations are algebraic equations in which the variable appears only to the first power and is not multiplied by itself. The general form is ax + b = c, where a, b, and c are constants.

• Linear Expression: An algebraic expression involving a variable to the first power (e.g., 2x + 3).

• Linear Equation: An equation that sets a linear expression equal to another value (e.g., x + 2 = 0).

• Operations: Addition, subtraction, multiplication, and division are used to isolate the variable.

Steps to Solve Linear Equations

1. Distribute constants across parentheses.

2. Combine like terms on each side.

3. Group terms with x and constants on opposite sides.

4. Isolate x (solve for x).

5. Check the solution by substituting x back into the original equation.

Linear Equations with Fractions

When fractions are present, multiply both sides by the Least Common Denominator (LCD) to eliminate them before proceeding with the standard steps.

Categorizing Linear Equations

• Conditional Equation: Has one solution (e.g., 2x + 4 = 10).

• Identity: True for all real numbers (e.g., x + 5 = x + 2 + 3).

• Inconsistent Equation: Has no solution (e.g., x = x + 4).

Solving Rational Equations

Definition and Restrictions

A rational equation contains a variable in the denominator of a fraction. Solutions must not make any denominator zero.

• Restriction: Set the denominator equal to zero to find values that are not allowed.

• Solving Steps:

  1. Determine restrictions by setting denominators to zero.

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

  3. Solve the resulting linear equation.

  4. Check solutions against restrictions.

The Imaginary Unit and Complex Numbers

Square Roots of Negative Numbers

Square roots of negative numbers are not real. The imaginary unit i is defined as .

• Imaginary Numbers: Numbers involving i (e.g., ).

Powers of i

Powers of i cycle every four:

Any higher power can be reduced to one of these four values using exponent properties.

Complex Numbers

A complex number is written in standard form as , where a is the real part and b is the imaginary part.

• Example: has real part 4 and imaginary part 3.

Operations with Complex Numbers

• Add/Subtract: Combine like terms (real with real, imaginary with imaginary).

• Multiply: Use distributive property or FOIL, simplify to -1.

• Complex Conjugate: The conjugate of is . Multiplying a complex number by its conjugate yields a real number: .

• Divide: Multiply numerator and denominator by the conjugate of the denominator to make the denominator real.

Quadratic Equations

Definition and Standard Form

A quadratic equation is a polynomial equation of degree 2, typically written as .

• Standard Form: All terms on one side, ordered by descending powers of x.

• Example: becomes .

Factoring Quadratic Equations

1. Write the equation in standard form.

2. Factor completely.

3. Set each factor equal to zero and solve for x.

4. Check solutions in the original equation.

The Square Root Property

Used when the equation can be written as . Take the square root of both sides and solve for x.

• Isolate the squared expression.

• Take both positive and negative square roots.

• Solve for x.

Completing the Square

Transform the equation into by adding a suitable constant to both sides.

1. Simplify to .

2. Add to both sides.

3. Factor left side as a perfect square.

4. Solve using the square root property.

The Quadratic Formula

The quadratic formula solves any quadratic equation:

• Plug in values for a, b, and c from the standard form.

• Compute and simplify solutions.

The Discriminant

The discriminant, , determines the number and type of solutions:

• If : Two real solutions

• If : One real solution

• If : Two complex (imaginary) solutions

Example: For , (one real solution).

Example: For , (two real solutions).

Choosing a Method to Solve Quadratic Equations

Depending on the form and coefficients, select the most efficient method:

• Factoring: Use if the equation has obvious factors or c = 0.

• Square Root Property: Use if the equation is in the form .

• Completing the Square: Use if b is even or leading coefficient is 1.

• Quadratic Formula: Use if factoring is difficult or unsure which method to use.

Linear Inequalities

Interval Notation

Interval notation is a compact way to express solution sets:

• Closed Interval: means .

• Open Interval: means .

• Half-Open Interval: or .

• Infinity: Use or for unbounded intervals, always with open bounds.

Solving Linear Inequalities

Linear inequalities are solved similarly to linear equations, but with an inequality symbol (e.g., , , , ).

• Apply the same steps as for linear equations.

• If you multiply or divide both sides by a negative number, reverse the inequality symbol.

• Express the solution set in interval notation.

Fractions and Variables on Both Sides

Solve inequalities with fractions and variables on both sides by clearing fractions and isolating the variable, just as with equations.

• Example:

Summary Table: Methods for Solving Quadratic Equations