SOLVING QUADRATIC EQUATIONS WITH RADICAL COEFFICIENTS


Note:




Solve for x in the following equation.

Example 3:tex2html_wrap_inline155tex2html_wrap_inline375

The equation is already set to zero.


Simplify the equation. tex2html_wrap_inline377


eqnarray50


eqnarray56







Method 1:tex2html_wrap_inline155Factoring

The equation tex2html_wrap_inline379 is not easily factored. Therefore, we will not use this method.







Method 2:tex2html_wrap_inline155Completing the square

Add 1 to both sides of the equation tex2html_wrap_inline381 .

eqnarray79



Divide both sides of the equation by tex2html_wrap_inline383 and simplify.


eqnarray88


eqnarray97


eqnarray106



Add tex2html_wrap_inline385 to both sides of the equation:


eqnarray123



Factor the left side and simplify the right side:


eqnarray145



Take the square root of both sides of the equation:


eqnarray163



Subtract tex2html_wrap_inline387 from both sides of the equation:


eqnarray180


eqnarray187



The exact answers are tex2html_wrap_inline389 and the approximate answers are tex2html_wrap_inline391







Method 3:tex2html_wrap_inline155Quadratic Formula

The quadratic formula is tex2html_wrap_inline393


In the equation tex2html_wrap_inline395 ,a is the coefficient of the tex2html_wrap_inline397 term, b is the coefficient of the x term, and c is the constant. Substitute tex2html_wrap_inline401 for a, tex2html_wrap_inline403 for b, and tex2html_wrap_inline405 for c in the quadratic formula and simplify.


eqnarray221


eqnarray229


eqnarray235



The exact answers are tex2html_wrap_inline389 and the approximate answers are tex2html_wrap_inline391







Method 4:tex2html_wrap_inline155Graphing

Graph the left side of the equation, tex2html_wrap_inline411 Graph tex2html_wrap_inline413 tex2html_wrap_inline415 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline417 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 0.351506755 and one at 1.6425012.


The answers are 0.351506755 and 1.6425012. 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=0.351506755 by substituting 0.351506755 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 0.351506755 for x, then x = 0.351506755 is a solution.


Check the solution x = 1.6425012 by substituting 1.6425012 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 1.6425012 for x, then x=1.6425012 is a solution.






The exact answers are tex2html_wrap_inline389 and the approximate answers are tex2html_wrap_inline391






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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