SOLVING QUADRATIC EQUATIONS

Note:


Solve for x in the following equation.

Example 4: tex2html_wrap_inline167

The equation is already equal to zero.





Method 1: tex2html_wrap_inline155 Factoring

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





Method 2: tex2html_wrap_inline155 Completing the square

Subtract 10 from both sides of the equation.

eqnarray45

Add tex2html_wrap_inline169 to both sides of the equation.

eqnarray52

Factor the left side and simplify the right side.

eqnarray56

Take the square root of both sides of the equation,

eqnarray60

Subtract 3 from both sides of the equation.

eqnarray64

and

eqnarray66





Method 3: tex2html_wrap_inline155 Quadratic Formula

The quadratic formula is tex2html_wrap_inline171 .

In the equation tex2html_wrap_inline167 , a is the coefficient of the tex2html_wrap_inline175 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.

eqnarray83

and

eqnarray93





Method 4: tex2html_wrap_inline155 Graphing

Graph y= the left side of the equation or tex2html_wrap_inline187 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_inline187 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 tex2html_wrap_inline203





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.

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.

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 tex2html_wrap_inline167 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 tex2html_wrap_inline241


Since x=-3+i, then tex2html_wrap_inline245

Since the product tex2html_wrap_inline247 and tex2html_wrap_inline167 , then we can say that tex2html_wrap_inline251 This means that tex2html_wrap_inline253 and tex2html_wrap_inline255 are factors of tex2html_wrap_inline257




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