Question 119412
#1




Let's use the quadratic formula to solve for x:



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{-2*x^2+x+8=0}}} ( notice {{{a=-2}}}, {{{b=1}}}, and {{{c=8}}})





{{{x = (-1 +- sqrt( (1)^2-4*-2*8 ))/(2*-2)}}} Plug in a=-2, b=1, and c=8




{{{x = (-1 +- sqrt( 1-4*-2*8 ))/(2*-2)}}} Square 1 to get 1  




{{{x = (-1 +- sqrt( 1+64 ))/(2*-2)}}} Multiply {{{-4*8*-2}}} to get {{{64}}}




{{{x = (-1 +- sqrt( 65 ))/(2*-2)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-1 +- sqrt(65))/(2*-2)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{x = (-1 +- sqrt(65))/-4}}} Multiply 2 and -2 to get -4


So now the expression breaks down into two parts


{{{x = (-1 + sqrt(65))/-4}}} or {{{x = (-1 - sqrt(65))/-4}}}



Now break up the fraction



{{{x=-1/-4+sqrt(65)/-4}}} or {{{x=-1/-4-sqrt(65)/-4}}}



Simplify



{{{x=1 / 4-sqrt(65)/4}}} or {{{x=1 / 4+sqrt(65)/4}}}



So these expressions approximate to


{{{x=-1.76556443707464}}} or {{{x=2.26556443707464}}}



So our solutions are:

{{{x=-1.76556443707464}}} or {{{x=2.26556443707464}}}


Notice when we graph {{{-2*x^2+x+8}}}, we get:


{{{ graph( 500, 500, -11.7655644370746, 12.2655644370746, -11.7655644370746, 12.2655644370746,-2*x^2+1*x+8) }}}


when we use the root finder feature on a calculator, we find that {{{x=-1.76556443707464}}} and {{{x=2.26556443707464}}}.So this verifies our answer



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#2


{{{x^2+18x+81=25}}} Start with the given equation



{{{x^2+18x+81-25=0}}} Subtract 25 from both sides



{{{x^2+18*x+56=0}}} Combine like terms




Let's use the quadratic formula to solve for x:



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{x^2+18*x+56=0}}} ( notice {{{a=1}}}, {{{b=18}}}, and {{{c=56}}})





{{{x = (-18 +- sqrt( (18)^2-4*1*56 ))/(2*1)}}} Plug in a=1, b=18, and c=56




{{{x = (-18 +- sqrt( 324-4*1*56 ))/(2*1)}}} Square 18 to get 324  




{{{x = (-18 +- sqrt( 324+-224 ))/(2*1)}}} Multiply {{{-4*56*1}}} to get {{{-224}}}




{{{x = (-18 +- sqrt( 100 ))/(2*1)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-18 +- 10)/(2*1)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{x = (-18 +- 10)/2}}} Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


{{{x = (-18 + 10)/2}}} or {{{x = (-18 - 10)/2}}}


Lets look at the first part:


{{{x=(-18 + 10)/2}}}


{{{x=-8/2}}} Add the terms in the numerator

{{{x=-4}}} Divide


So one answer is

{{{x=-4}}}




Now lets look at the second part:


{{{x=(-18 - 10)/2}}}


{{{x=-28/2}}} Subtract the terms in the numerator

{{{x=-14}}} Divide


So another answer is

{{{x=-14}}}


So our solutions are:

{{{x=-4}}} or {{{x=-14}}}


Notice when we graph {{{x^2+18*x+56}}}, we get:


{{{ graph( 500, 500, -24, 6, -24, 6,1*x^2+18*x+56) }}}


and we can see that the roots are {{{x=-4}}} and {{{x=-14}}}. This verifies our answer




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*[invoke simplifying_square_roots -175]