SOLVING QUADRATIC EQUATIONS

Note:




Solve for x in the following equation.

Example 2:

tex2html_wrap_inline222

Set the equation equal to zero by subtracting 3x and 10 from both sides of the equation.

eqnarray35







Method 1:Factoring

eqnarray43







Method 2:Completing the square


Add 7 to both sides of the equation tex2html_wrap_inline228

eqnarray57



Divide both sides by 12 :

eqnarray62



Add tex2html_wrap_inline230 to both sides of the equation:

eqnarray77



Factor the left side and simplify the right side :

eqnarray90



Take the square root of both sides of the equation :

eqnarray98



Add tex2html_wrap_inline232 to both sides of the equation :

eqnarray107







Method 3:Quadratic Formula

The quadratic formula is tex2html_wrap_inline234.



In the equation tex2html_wrap_inline236 , a is the coefficient of the tex2html_wrap_inline238 term, b is the coefficient of the x term, and c is the constant.Substitute 12 for a, -25 for b , and -7 for c in the quadratic formula and simplify.

eqnarray145

eqnarray152







Method 4: Graphing

Graph tex2html_wrap_inline248 (This equation is formed by subtracting the right side of the original graph from the left side of the original graph.) Graph tex2html_wrap_inline250 The graph of tex2html_wrap_inline252 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline254 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 2.333333 and -0.25. This means that there are two real answers: x=2.333333 and -0.25.

The answers are 2.333333 and -0.25. 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=2.333333 by substituting 2.333333 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 x = 2.333333 for x, then x = 2.333333 is a solution.





Check the solution x=-0.25 by substituting -0.25 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 -0.25 for x, then x=-0.25 is a solution.





The solutions to the equation tex2html_wrap_inline222 are - 0.25 and 2.333333.








If you would like to work another example, click on Example.


If you would like to test yourself by working some problems similar to this example, click on Problem.


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Author: Nancy Marcus

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