SOLVING QUADRATIC EQUATIONS WITH RADICAL COEFFICIENTS


Note:




Solve for x in the following equation.

Example 1:tex2html_wrap_inline155tex2html_wrap_inline346


The equation is already set to zero.







Method 1:tex2html_wrap_inline155Factoring

The equation tex2html_wrap_inline348 is not easily factored. Therefore, we will not use this method.







Method 2:tex2html_wrap_inline155Completing the square


Divide both sides of the equation by tex2html_wrap_inline350


eqnarray57


eqnarray66


eqnarray76



Add tex2html_wrap_inline352 to both sides of the equation:


eqnarray96



Factor the left side and simplify the right side:


eqnarray115



Take the square root of both sides of the equation :


eqnarray137



Subtract tex2html_wrap_inline354 from both sides of the equation:


eqnarray150


eqnarray157



The exact answers are tex2html_wrap_inline356 and the approximate answers are tex2html_wrap_inline358 2.0646448.







Method 3:tex2html_wrap_inline155Quadratic Formula


The quadratic formula is tex2html_wrap_inline362


In the equation tex2html_wrap_inline364 ,a is the coefficient of the tex2html_wrap_inline366 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline370 for a, tex2html_wrap_inline372 for b, and tex2html_wrap_inline374 for c in the quadratic formula and simplify.



eqnarray188


eqnarray196


eqnarray203



The exact answers are tex2html_wrap_inline356 and the approximate answers are tex2html_wrap_inline358 2.0646448 .







Method 4:tex2html_wrap_inline155Graphing


Graph the left side of the equation, tex2html_wrap_inline382 (formed by subtracting the right side of the original equation from the left side of the original equation).Graph tex2html_wrap_inline386 tex2html_wrap_inline386 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline390 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 -3.355639 and one at 2.0646448.


The answers are -3.355639 and 2.0646448. 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=-3.355639 by substituting -3.355639 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 -3.355639 for x, then x=-3.355639 is a solution.


Check the solution x=2.0646448 by substituting -2.0646448 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 2.0646448 for x, then x=2.0646448 is a solution.







The exact answers are tex2html_wrap_inline356 and the approximate answers are tex2html_wrap_inline434








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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