SOLVING QUADRATIC EQUATIONS


Note:




Solve for x in the following equation.


Problem 4.3c: tex2html_wrap_inline155 tex2html_wrap_inline468





Answer: tex2html_wrap_inline155 tex2html_wrap_inline470 exact answers;


tex2html_wrap_inline472 and tex2html_wrap_inline474 approximate answers







Solution:


The equation is already set to zero. Simplify the equation to tex2html_wrap_inline476 .







Method 1: tex2html_wrap_inline155 Factoring


The equation tex2html_wrap_inline478 is not easily factored, so we will not use this method.







Method 2: tex2html_wrap_inline155 Completing the square


Add 13 to both sides of the equation tex2html_wrap_inline478 .


eqnarray53


Divide both sides by 4:


eqnarray65


Add tex2html_wrap_inline482 to both sides of the equation:


eqnarray88


Factor the left side and simplify the right side :


eqnarray107


Take the square root of both sides of the equation:


eqnarray122


Add tex2html_wrap_inline484 to both sides of the equation:


eqnarray140


eqnarray149







Method 3: tex2html_wrap_inline155 Quadratic Formula


The quadratic formula is tex2html_wrap_inline486


In the equation tex2html_wrap_inline488 ,a is the coefficient of the tex2html_wrap_inline490 term, b is the coefficient of the x term, and c is the constant. Substitute 4 for a, -7 for b, and -13 for c in the quadratic formula and simplify.


eqnarray180


eqnarray189


eqnarray196







Method 4: tex2html_wrap_inline155 Graphing


Graph tex2html_wrap_inline500 (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_inline500 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 2.87290244274 and -1.12890244274. This means that there are two real answers: x=2.87290244274 and tex2html_wrap_inline514


The answers are 2.87290244274 and -1.12890244274. 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 x=2.87290244274 by substituting 2.87290244274 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 2.87290244274 for x, then x=2.87290244274 is a solution.



Check the solution x=-1.12890244274 by substituting -1.12890244274 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 -1.12890244274 for x, then x=-1.12890244274 is a solution.







The solutions to the equation tex2html_wrap_inline476 are tex2html_wrap_inline552 and tex2html_wrap_inline554







Comment: tex2html_wrap_inline155 You can use the exact solutions to factor the original equation tex2html_wrap_inline476 .


For example, since tex2html_wrap_inline558 , then


eqnarray250


Since tex2html_wrap_inline564 , then


eqnarray259


Since the product tex2html_wrap_inline568


and tex2html_wrap_inline570 then tex2html_wrap_inline572 and tex2html_wrap_inline574 are factors of tex2html_wrap_inline576


However not the only factors:


Since the first term of the product tex2html_wrap_inline578


is not tex2html_wrap_inline580 there must be another factor of 4:


Let tex2html_wrap_inline582 s check to see whether tex2html_wrap_inline584


tex2html_wrap_inline586


displaymath464


displaymath465


eqnarray362


eqnarray383


Therefore tex2html_wrap_inline588 is factored as tex2html_wrap_inline590






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


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Author: Nancy Marcus

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