SOLVING QUADRATIC EQUATIONS



Note:




Solve for x in the following equation.


Problem 4.3a:text2html_wrap_inline253tex2html_wrap_inline344


Answer:text2html_wrap_inline253tex2html_wrap_inline346


Solution:

The equation is already set to zero.







Method 1:text2html_wrap_inline253Factoring

The equation tex2html_wrap_inline348 can be written as

eqnarray47



The only way a product can equal zero is for at least one of the factors to have a value of zero :

If 3x-4 = 0, then tex2html_wrap_inline354

If 2x+5 = 0, then tex2html_wrap_inline358







Method 2:text2html_wrap_inline253Completing the square

Add 20 to both sides of the equation tex2html_wrap_inline348 .


eqnarray72



Divide both sides by 6:

eqnarray85



Add tex2html_wrap_inline362 to both sides of the equation:

eqnarray108



Factor the left side and simplify the right side:

eqnarray127



Take the square root of both sides of the equation:

eqnarray143



Subtract tex2html_wrap_inline366 from both sides of the equation:

eqnarray162


eqnarray172







Method 3:text2html_wrap_inline253Quadratic Formula

The quadratic formula is tex2html_wrap_inline370

In the equation tex2html_wrap_inline344 , a is the coefficient of the tex2html_wrap_inline374 term, b is the coefficient of the x term, and c is the constant. Substitute 6 for a, 7 for b, and -20 for c in the quadratic formula and simplify.



eqnarray219


eqnarray229


eqnarray236







Method 4:text2html_wrap_inline253Graphing

Graph tex2html_wrap_inline384 (formed by subtracting the right side of the original equation from the left side of the original equation). Graph y=0 (the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline384 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 1.333333 and -2.5. This means that there are two real answers: x=1.33333 and tex2html_wrap_inline400

The answers are tex2html_wrap_inline402 and tex2html_wrap_inline404 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 tex2html_wrap_inline354 by substituting tex2html_wrap_inline402 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 tex2html_wrap_inline402 for x, then tex2html_wrap_inline354 is a solution.




Check the solution tex2html_wrap_inline358 by substituting tex2html_wrap_inline428 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 tex2html_wrap_inline428 for x, then tex2html_wrap_inline358 is a solution.





The solutions to the equation tex2html_wrap_inline344 are tex2html_wrap_inline402 and tex2html_wrap_inline404






If you would like to review the solution to 4.3b, click on Problem


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Author:Nancy Marcus

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