SOLVING QUADRATIC EQUATIONS

Note:


Solve for x in the following equation.

Example 2: tex2html_wrap_inline253 tex2html_wrap_inline253

The equation is already equal to zero.




Method 1:tex2html_wrap_inline253 Factoring

The left side of the equation is not easily factored, so we will not use this method.





Method 2:tex2html_wrap_inline253 Completing the square


Subtract 3 from both sides of the equation.

eqnarray38




Add tex2html_wrap_inline255 to both sides of the equation:

eqnarray48




Factor the left side and simplify the right side.

eqnarray57




Take the square root of both sides of the equation.

eqnarray67




Add tex2html_wrap_inline257 to both sides of the equation.

eqnarray77

and

eqnarray77








Method 3:tex2html_wrap_inline253 Quadratic Formula

The quadratic formula is tex2html_wrap_inline259

In the equation tex2html_wrap_inline253 , a is the coefficient of the tex2html_wrap_inline263 term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, -5 for b, and 3 for c in the quadratic formula and simplify

.

eqnarray107

and

eqnarray107








Method 4:tex2html_wrap_inline253 Graphing

Graph y= the left side of the equation or tex2html_wrap_inline275 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_inline275 crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation. The x-intercepts are 4.30277563773 and tex2html_wrap_inline289

The exact answers are tex2html_wrap_inline291 and tex2html_wrap_inline293 The approximate answers are 4.30277563773 and tex2html_wrap_inline299



Check these answers in the original equation.


Check the solution x=4.30277563773 by substituting 4.30277563773 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 4.30277563773 for x, then x=4.30277563773 is a solution.





Check the solution x=0.697224362268 by substituting 0.697224362268 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 0.697224362268 for x, then x=0.697224362268 is a solution.




The solutions to the equation tex2html_wrap_inline253 tex2html_wrap_inline253 tex2html_wrap_inline253 are tex2html_wrap_inline253 tex2html_wrap_inline291 tex2html_wrap_inline253 and tex2html_wrap_inline253 tex2html_wrap_inline331 tex2html_wrap_inline253 rounded to 4.30277563773 and 0.697224362268





Comment: tex2html_wrap_inline253 You can use the solutions to factor the original equation.


For example, since tex2html_wrap_inline337 , then tex2html_wrap_inline339 , and tex2html_wrap_inline341

Since tex2html_wrap_inline343 , then tex2html_wrap_inline345 , and tex2html_wrap_inline347

Since the product tex2html_wrap_inline349 and tex2html_wrap_inline253 , then we can say that tex2html_wrap_inline353 This means that tex2html_wrap_inline355 and tex2html_wrap_inline357
are factors of tex2html_wrap_inline359





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