SOLVING QUADRATIC EQUATIONS

Note:

• A quadratic equation is a polynomial equation of degree 2.

• The ''U'' shaped graph of a quadratic is called a parabola.

• A quadratic equation has two solutions. Either two distinct real solutions, one double real solution or two imaginary solutions.

• There are several methods you can use to solve a quadratic equation:
  1. Factoring
  2. Completing the Square
  3. Quadratic Formula
  4. Graphing

• All methods start with setting the equation equal to zero.

Solve for x in the following equation.

Example 4:

The equation is already equal to zero.

Method 1: Factoring

We will not use this method because the left side of the equation is not easily factored.

Method 2: Completing the square

Subtract 10 from both sides of the equation.

Add to both sides of the equation.

Factor the left side and simplify the right side.

Take the square root of both sides of the equation,

Subtract 3 from both sides of the equation.

and

Method 3: Quadratic Formula

The quadratic formula is .

In the equation , a is the coefficient of the term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, 6 for b, and 10 for c in the quadratic formula and simplify.

and

Method 4: Graphing

Graph y= the left side of the equation or and graph y= the right side of the equation or y=0. The graph of y=0 is nothing more than the x-axis. So what you will be looking for is where the graph of crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation. There are no x-intercepts. This means that there are no real solutions; the only solutions will be imaginary. The answers are -3+i and

Check these answers in the original equation.

Check the answer x=-3+i by substituting -3+i in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

  

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value -3+i for x, then x=-3+i is a solution.

Check the solution x=-3-i by substituting -3-i in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

  

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value -3-i for x, then x=-3-i is a solution.

The solutions to the equation are x = - 3 - i and - 3 + i.

Comment: You can use the solutions to factor the original equation.

For example, since x=-3-i, then

Since x=-3+i, then

Since the product and , then we can say that This means that and are factors of

If you would like to go work another example, click on Example

If you would like to test yourself by working some problems similar to this example, click on Problem.

If you would like to go back to the equation table of contents, click on Contents.

[Algebra] [Trigonometry] [Geometry] [Differential Equations] [Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Copyright © 1999-2024 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour