SOLVING QUADRATIC EQUATIONS

Note:


Solve for x in the following equation.

Example 3: tex2html_wrap_inline155 tex2html_wrap_inline139

The equation is already set to zero.





Method 1: tex2html_wrap_inline155 Factoring

The equation can be written in the equivalent form of tex2html_wrap_inline141 The only way that a product can equal zero is if one or both of the factors equal zero.

eqnarray43

Both answers are 8. This means that 8 is a double zero or double solution.





Method 2: tex2html_wrap_inline155 Completing the square

Subtract 64 from both sides of the equation.

eqnarray50

Add tex2html_wrap_inline143 to both sides of the equation.

eqnarray57

Factor the left side and simplify the right side.

eqnarray61

Take the square root of both sides of the equation,

eqnarray65

Add 8 to both sides of the equation.

eqnarray68





Method 3: tex2html_wrap_inline155 Quadratic Formula

The quadratic formula is tex2html_wrap_inline145

In the equation tex2html_wrap_inline139 , a is the coefficient of the tex2html_wrap_inline149 term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, -16 for b, and 64 for c in the quadratic formula and simplify

.

eqnarray85





Method 4: tex2html_wrap_inline155 Graphing

Graph y= the left side of the equation or tex2html_wrap_inline161 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 tex2html_wrap_inline169 crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation.

The only x-intercept is located at x=8. When the graph is tangent to the x-axis at x=8, it means that there is a double zero (or solution or x-intercept) at tex2html_wrap_inline177 The answer is tex2html_wrap_inline179





Check these answer in the original equation.

Check the answer x=8 by substituting 8 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.

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 8 for x, then x=8 is a solution.

The solution to the equation tex2html_wrap_inline139 is 8.




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.

Author:Nancy Marcus

Copyright 1999-2017 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