SOLVING QUADRATIC EQUATIONS

Note:


Solve for x in the following equation.

Example 5: tex2html_wrap_inline360

Set the equation equal to zero by subtracting tex2html_wrap_inline362 and adding tex2html_wrap_inline364 to sides of the equation.

eqnarray27

eqnarray37

eqnarray47

eqnarray57

Method 1: Factoring

The expression tex2html_wrap_inline366 cannot easily be factored, so we will not use this method.





Method 2: Completing the square

Divide both sides of the equation tex2html_wrap_inline368 by 24.

eqnarray72

eqnarray81

Add tex2html_wrap_inline370 to both sides of the equation:

eqnarray102

Factor the left side and simplify the right side :

eqnarray114

Take the square root of both sides of the equation :

eqnarray122

Subtract tex2html_wrap_inline372 from both sides of the equation :

eqnarray131

and

eqnarray141





Method 3: Quadratic Formula

The quadratic formula is tex2html_wrap_inline374

In the equation tex2html_wrap_inline368 , a is the coefficient of the tex2html_wrap_inline378 term, b is the coefficient of the x term, and c is the constant. Simply insert 24 for a, 33 for b, and -136 for c in the quadratic formula and simplify.

eqnarray163

eqnarray168

eqnarray175

and

eqnarray179





Method 4:Graphing

Graph tex2html_wrap_inline382 . The equation represents the left side of the original equation minus the right side of the original equation. The right side of the equation is now zero. 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_inline382 crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation.

You can see from the graph that there are two x-intercepts located at -3.165266 and 1.790266. This means that there are two real answers: x = - 3.165266 and 1.790266 .

The answers are -3.165266 and 1.790266.





These answers may or may not be the solutions to the original equation. Check these answers in the original equation.





Check the solution x = - 3.165266 by substituting - 3.165266 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 - 3.165266 for x, then x = - 3.165266 is a solution.





Check the solution x = 1.790260 by substituting 1.790260 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 1.790260 for x, then x=1.790260 is a solution.




The solutions to the equation tex2html_wrap_inline360 are -3.165266 and 1.790260.



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

For example, since tex2html_wrap_inline430 , then

eqnarray263

Since tex2html_wrap_inline434 , then

eqnarray276

Since the product

eqnarray287

then we can say that

eqnarray298

This means that tex2html_wrap_inline438 and tex2html_wrap_inline440 are factors of tex2html_wrap_inline442



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