SOLVING QUADRATIC EQUATIONS WITH RADICAL COEFFICIENTS


Note:




Solve for x in the following equation.

Example 2:tex2html_wrap_inline155tex2html_wrap_inline368

The equation is already set to zero.


Simplify the equation.


eqnarray47



can be rewritten as:


eqnarray54







Method 1:tex2html_wrap_inline155Factoring

The equation tex2html_wrap_inline372 is not easily factored. Therefore, we will not use this method.







Method 2:tex2html_wrap_inline155Completing the square


Add 8 to both sides of the equation tex2html_wrap_inline374 .


eqnarray74



Divide both sides of the equation tex2html_wrap_inline376 by tex2html_wrap_inline378 and simplify.


eqnarray85


eqnarray96


eqnarray106



Add tex2html_wrap_inline380 to both sides of the equation:


eqnarray134



Factor the left side and simplify the right side:


eqnarray152



Take the square root of both sides of the equation:


eqnarray168



Subtract tex2html_wrap_inline382 from both sides of the equation:


eqnarray185


eqnarray192


and


eqnarray199



The exact answers are tex2html_wrap_inline384 and the approximate answers are tex2html_wrap_inline386







Method 3:text2html_wrap_inline155 Quadratic Formula


The quadratic formula is tex2html_wrap_inline388


In the equation tex2html_wrap_inline390 , a is the coefficient of the tex2html_wrap_inline392 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline396 for a, tex2html_wrap_inline398 for b, and -8 for c in the quadratic formula and simplify.


eqnarray229


eqnarray237


eqnarray243



The exact answers are tex2html_wrap_inline402 and the approximate answers are tex2html_wrap_inline404







Method 4:tex2html_wrap_inline155Graphing


Graph the left side of the equation, tex2html_wrap_inline406 Graph tex2html_wrap_inline408 y=0 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline412 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, one at 0.77292396 and one at -2.439590637.


The answers are 0.77292396 and -2.43950637. 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 x=0.77292396 by substituting 0.77292396 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 0.77292396 for x, then x=0.77292396 is a solution.


Check the solution x=-2.439590637 by substituting -2.439590637 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 -2.439590637 for x, then x=-2.439590637 is a solution.







The exact answers are tex2html_wrap_inline402 and the approximate answers are tex2html_wrap_inline404








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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