Question 115684
#1

Well you're forgetting the other part of the answer which is {{{x=-sqrt(3)}}}. You are correct, there should be two answers.


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#2


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^2-14*x-10=0}}} ( notice {{{a=2}}}, {{{b=-14}}}, and {{{c=-10}}})





{{{x = (--14 +- sqrt( (-14)^2-4*2*-10 ))/(2*2)}}} Plug in a=2, b=-14, and c=-10




{{{x = (14 +- sqrt( (-14)^2-4*2*-10 ))/(2*2)}}} Negate -14 to get 14




{{{x = (14 +- sqrt( 196-4*2*-10 ))/(2*2)}}} Square -14 to get 196  (note: remember when you square -14, you must square the negative as well. This is because {{{(-14)^2=-14*-14=196}}}.)




{{{x = (14 +- sqrt( 196+80 ))/(2*2)}}} Multiply {{{-4*-10*2}}} to get {{{80}}}




{{{x = (14 +- sqrt( 276 ))/(2*2)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (14 +- 2*sqrt(69))/(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 = (14 +- 2*sqrt(69))/4}}} Multiply 2 and 2 to get 4


So now the expression breaks down into two parts


{{{x = (14 + 2*sqrt(69))/4}}} or {{{x = (14 - 2*sqrt(69))/4}}}



Now break up the fraction



{{{x=+14/4+2*sqrt(69)/4}}} or {{{x=+14/4-2*sqrt(69)/4}}}



Simplify



{{{x=7 / 2+sqrt(69)/2}}} or {{{x=7 / 2-sqrt(69)/2}}}



So these expressions approximate to


{{{x=7.65331193145904}}} or {{{x=-0.653311931459037}}}



So our solutions are:

{{{x=7.65331193145904}}} or {{{x=-0.653311931459037}}}


Notice when we graph {{{2*x^2-14*x-10}}}, we get:


{{{ graph( 500, 500, -10.653311931459, 17.653311931459, -10.653311931459, 17.653311931459,2*x^2+-14*x+-10) }}}


when we use the root finder feature on a calculator, we find that {{{x=7.65331193145904}}} and {{{x=-0.653311931459037}}}.So this verifies our answer



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{{{-2u^2+6=3u^2-10u}}} Start with the given equation



{{{-2u^2+6-3u^2+10u=0}}} Move all of the terms to the left side



{{{-2u^2-3u^2+10u+6=0}}} Sort the terms



{{{-5u^2+10u+6=0}}} Combine like terms




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



Starting with the general quadratic


{{{au^2+bu+c=0}}}


the general solution using the quadratic equation is:


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




So lets solve {{{-5*u^2+10*u+6=0}}} ( notice {{{a=-5}}}, {{{b=10}}}, and {{{c=6}}})





{{{u = (-10 +- sqrt( (10)^2-4*-5*6 ))/(2*-5)}}} Plug in a=-5, b=10, and c=6




{{{u = (-10 +- sqrt( 100-4*-5*6 ))/(2*-5)}}} Square 10 to get 100  




{{{u = (-10 +- sqrt( 100+120 ))/(2*-5)}}} Multiply {{{-4*6*-5}}} to get {{{120}}}




{{{u = (-10 +- sqrt( 220 ))/(2*-5)}}} Combine like terms in the radicand (everything under the square root)




{{{u = (-10 +- 2*sqrt(55))/(2*-5)}}} 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>)




{{{u = (-10 +- 2*sqrt(55))/-10}}} Multiply 2 and -5 to get -10


So now the expression breaks down into two parts


{{{u = (-10 + 2*sqrt(55))/-10}}} or {{{u = (-10 - 2*sqrt(55))/-10}}}



Now break up the fraction



{{{u=-10/-10+2*sqrt(55)/-10}}} or {{{u=-10/-10-2*sqrt(55)/-10}}}



Simplify



{{{u=1-sqrt(55)/5}}} or {{{u=1+sqrt(55)/5}}}



So these expressions approximate to


{{{u=-0.483239697419133}}} or {{{u=2.48323969741913}}}



So our solutions are:

{{{u=-0.483239697419133}}} or {{{u=2.48323969741913}}}


Notice when we graph {{{-5*x^2+10*x+6}}} (just replace u with x), we get:


{{{ graph( 500, 500, -10.4832396974191, 12.4832396974191, -10.4832396974191, 12.4832396974191,-5*x^2+10*x+6) }}}


when we use the root finder feature on a calculator, we find that {{{x=-0.483239697419133}}} and {{{x=2.48323969741913}}}.So this verifies our answer