SOLVING QUADRATIC EQUATIONS

Note:




Solve for x in the following equation.



Example 3:

tex2html_wrap_inline196

The equation is already equal to zero.







Method 1:text2html_wrap_inline253Factoring

Since the equation is not easily factored, we will skip this method.







Method 2:text2html_wrap_inline253Completing the square

Add 4 to both sides of the equation tex2html_wrap_inline198 .

eqnarray38



Divide both sides by 7 :

eqnarray43



Add tex2html_wrap_inline200 to both sides of the equation :

eqnarray58



Factor the left side and simplify the right side :

eqnarray70



Take the square root of both sides of the equation :

eqnarray78



Add tex2html_wrap_inline202 to both sides of the equation :

eqnarray87







Method 3:text2html_wrap_inline253Quadratic Formula

The quadratic formula is tex2html_wrap_inline204

In the equation tex2html_wrap_inline196 , a is the coefficient of the tex2html_wrap_inline208 term, b is the coefficient of the x term, and c is the constant. Substitute 7 for a, -5 for b, and -4 for c in the quadratic formula and simplify

.

eqnarray116

eqnarray123







Method 4:text2html_wrap_inline253Graphing

Graph tex2html_wrap_inline218 and 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_inline218 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 1.193193 and -0.478907. This means that there are two real answers: x=1.193193 and tex2html_wrap_inline236 The answers are 1.193193 and -0.478907. 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=1.193193 by substituting 1.193193 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=1.193193 for x, then x=1.193193 is a solution.





Check the solution x=-0.478907 by substituting -0.478907 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 -0.478907 for x, then x=-0.478907 is a solution.





The solutions to the equation tex2html_wrap_inline196 are - 0.478907 and 1.193193.






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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