Algebraic Equations and Expressions

Solving Radical Equations

Radical equations are equations in which the variable appears under a radical sign. To solve these, isolate the radical and then raise both sides to the appropriate power to eliminate the radical.

• Key Point: Always check for extraneous solutions after solving, as squaring both sides can introduce invalid solutions.

• Example: Solve .

  1. Isolate one radical:

  2. Square both sides:

  3. Expand and solve for .

Solving Polynomial Equations

Polynomial equations involve variables raised to whole number powers. Solutions can be found by factoring, using the quadratic formula, or other algebraic methods.

• Key Point: For higher-degree polynomials, look for factoring opportunities or use substitution.

• Example: Solve .

  • Let , so the equation becomes .

  • Factor:

  • So or , thus or .

  • Therefore, or .

Solving Inequalities and Interval Notation

Inequalities are solved similarly to equations, but the solution is a range of values. Interval notation is used to express these solutions compactly.

• Key Point: When multiplying or dividing both sides by a negative number, reverse the inequality sign.

• Example: Solve .

  • Isolate the absolute value:

  • Write as a compound inequality:

  • Solve for :

  • Interval notation:

Complex Numbers and Standard Form

Writing Expressions in Standard Form

Complex numbers are written in the form , where and are real numbers and is the imaginary unit ().

• Key Point: Combine like terms and simplify using .

• Examples:

  • a) Combine real and imaginary parts: (real), (imaginary) Standard form:

  • b) Multiply numerator and denominator by the conjugate :

  • c) Expand:

Quadratic Equations

Solving by Completing the Square

Completing the square is a method to solve quadratic equations by rewriting them in the form .

• Key Point: Move constant to the other side, add to both sides, then solve for .

• Example: Solve by completing the square.

  1. Move constant:

  2. Add to both sides:

  3. Write as square:

  4. Solve: , so

Solving by Quadratic Formula

The quadratic formula solves any quadratic equation :

• Formula:

• Example: Solve .

  • , ,

  • Discriminant:

  • So or

Graphing Quadratic Functions

Plotting Points and Analyzing Symmetry

Quadratic functions have the general form . Their graphs are parabolas, which may open upward or downward depending on the sign of .

• Key Point: The axis of symmetry is .

• Intercepts:

  • Y-intercept: Set , solve for .

  • X-intercepts: Set , solve for .

• Example: Graph by plotting points.

  • Vertex:

  • Axis of symmetry:

  • Y-intercept:

  • X-intercepts:

  • Plot points for and connect to form a downward-opening parabola.

Summary Table: Quadratic Equation Methods