SOLVING QUADRATIC EQUATIONS WITH RADICAL COEFFICIENTS


Note:




Solve for x in the following equation.

Example 4:tex2html_wrap_inline155 tex2html_wrap_inline409

The equation is already set to zero.


Simplify the equation. tex2html_wrap_inline411


eqnarray53


eqnarray64



eqnarray76


eqnarray84







Method 1:tex2html_wrap_inline155Factoring

The equation tex2html_wrap_inline413 is not easily factored. Therefore, we will not use this method.







Method 2:tex2html_wrap_inline155Completing the square

Add tex2html_wrap_inline415 to both sides of the equation tex2html_wrap_inline417 .


eqnarray106



Divide both sides of the equation by tex2html_wrap_inline419 and simplify.


eqnarray114


eqnarray123


eqnarray133



Add tex2html_wrap_inline421 to both sides of the equation:


eqnarray151



Factor the left side and simplify the right side:


eqnarray169



Take the square root of both sides of the equation:


eqnarray183



Subtract tex2html_wrap_inline423 from both sides of the equation:


eqnarray198


eqnarray205



The exact answers are tex2html_wrap_inline425 and the approximate answers are tex2html_wrap_inline427







Method 3:tex2html_wrap_inline155Quadratic Formula

The quadratic formula is tex2html_wrap_inline429


In the equation tex2html_wrap_inline431 ,a is the coefficient of the tex2html_wrap_inline433 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline437 for a, tex2html_wrap_inline439 for b, and tex2html_wrap_inline441 for c in the quadratic formula and simplify.


eqnarray239


eqnarray246


eqnarray251



The exact answers are tex2html_wrap_inline443 and the approximate answers are tex2html_wrap_inline445







Method 4:tex2html_wrap_inline155Graphing


Graph the left side of the equation, tex2html_wrap_inline447 Graph tex2html_wrap_inline449 tex2html_wrap_inline451 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline453 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 2.7616977 and one at -3.710381013.


The answers are 2.7616977 and -3.710381013. 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=2.7616977 by substituting 2.7616977 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 2.7616977 for x, then x=2.7616977 is a solution.


Check the solution x=-3.710381013 by substituting -3.710381013 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 -3.710381013 for x, then x=-3.710381013 is a solution.






The exact answers are tex2html_wrap_inline495 and the approximate answers are tex2html_wrap_inline445







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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