SOLVING QUADRATIC EQUATIONS


Note:




Solve for x in the following equation.


Example 1: text2html_wrap_inline253 tex2html_wrap_inline443

Set the equation equal to zero by subtracting 3 x and 7 from both sides of the equation.


eqnarray17


eqnarray21







Method 1:text2html_wrap_inline253 Factoring

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







Method 2:text2html_wrap_inline253 Completing the Square

Subtract 15 from both sides of the equation tex2html_wrap_inline451 .


eqnarray40



Add tex2html_wrap_inline453 to both sides of the equation:


eqnarray51



Factor the left side and simplify the right side :


eqnarray61



Take the square root of both sides of the equation:


eqnarray69



Add tex2html_wrap_inline455 to both sides of the equation:


eqnarray83


and


eqnarray93









Method 3:text2html_wrap_inline253 Quadratic Formula


The quadratic formula is tex2html_wrap_inline457


In the equation tex2html_wrap_inline459 ,a is the coefficient of the tex2html_wrap_inline461 term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, -3 for b, and 15 for c in the quadratic formula and simplify.




eqnarray115


eqnarray119


eqnarray125


and


eqnarray129







Method 4:text2html_wrap_inline253 Graphing

Graph y= the left side of the equation or tex2html_wrap_inline467 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_inline467 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 no x-intercepts. This means that there are no real answers; the solution are two imaginary numbers.


The answers are tex2html_wrap_inline481 and tex2html_wrap_inline483 These answers may or may not be solutions to the original equations. You must verify that these answers are solutions.



Check these answers in the original equation.



Check the solution tex2html_wrap_inline491 by substituting tex2html_wrap_inline481 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 tex2html_wrap_inline481 for x, then tex2html_wrap_inline491 is a solution.



Check the solution tex2html_wrap_inline503 by substituting tex2html_wrap_inline505 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 tex2html_wrap_inline505 for x, then tex2html_wrap_inline503 is a solution.







The solutions to the equation tex2html_wrap_inline451 are tex2html_wrap_inline481 and tex2html_wrap_inline483







Comment: text2html_wrap_inline253 You can use the solutions to factor the original equation.


For example, since tex2html_wrap_inline521 ,then


eqnarray306


Since tex2html_wrap_inline527 , then


eqnarray319


tex2html_wrap_inline529


Since the product


eqnarray330 then we can say that


eqnarray341



This means that tex2html_wrap_inline531 and tex2html_wrap_inline533 are factors of tex2html_wrap_inline535








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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