SOLVING QUADRATIC EQUATIONS

Note:


Solve for x in the following equation.

Problem 4.2c:

tex2html_wrap_inline226



Answer: x = -2, -9





Solution:

Set the equation tex2html_wrap_inline230 equal to zero by subtracting 6x and adding 8 to both sides of the equation.

eqnarray51

eqnarray55

eqnarray59




Method 1:Factoring

The equation tex2html_wrap_inline236 can be written as

eqnarray66

The only way a product can equal zero is for aat least one of the factors to have a value of zero:

eqnarray71




Method 2:Completing the square

Subtract 18 from both sides of the equation tex2html_wrap_inline236 .

eqnarray80

Add tex2html_wrap_inline244 to both sides of the equation:

eqnarray90

Factor the left side and simplify the right side :

eqnarray98

Take the square root of both sides of the equation :

eqnarray106

Subtract tex2html_wrap_inline246 from both sides of the equation :

eqnarray115

and

eqnarray127





Method 3:Quadratic Formula

The quadratic formula is tex2html_wrap_inline248

In the equation tex2html_wrap_inline250 , a is the coefficient of the tex2html_wrap_inline252 term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, 11 for b, and 18 for c in the quadratic formula and simplify.

eqnarray150

eqnarray157




Method 4:Graphing

Graph tex2html_wrap_inline256 (formed by subtracting the right side of the original equation from the left side of the original equation. Graph y = 0 (the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline260 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 -2 and -9. This means that there are two real answers: x = -2 and tex2html_wrap_inline272

The answers are x=-2 and -9. These answers may or may not be solutions to the original equation. You must check the answers with the original equation.




Check these answers in the original equation.




Check the solution x=-2 by substituting -2 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 -2 for x, then x = -2 is a solution.




Check the solution x = -9 by substituting -9 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 -9 for x, then x=-9 is a solution.




The solutions to the equation tex2html_wrap_inline230 are -2 and tex2html_wrap_inline312



If you would like to review the solution to 4.2d, click on Solution.

If you would like to go back to the problem page, 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