Question 120450
#1



{{{5x^2=3x+2}}} Start with the given equation



{{{0=3x+2-5x^2}}}  Subtract 5x^2 from both sides. 



{{{0=-5x^2+3x+2}}}  Rearrange the terms. 



{{{0=-(x-1)(5x+2)}}} Factor the right side (note: if you need help with factoring, check out this <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)




Now set each factor equal to zero:

{{{x-1=0}}} or  {{{5x+2=0}}} 


{{{x=1}}} or  {{{x=-2/5}}}    Now solve for x in each case



So our answer is 

 {{{x=1}}} or  {{{x=-2/5}}} 



Notice if we graph {{{y=-5x^2+3x+2}}}  we can see that the roots are {{{x=1}}} and  {{{x=-2/5}}} . So this visually verifies our answer.



{{{ graph(500,500,-10,10,-10,10,0, -5x^2+3x+2) }}}



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#2



{{{3x^3+2x^2=x}}} Start with the given equation



{{{3x^3+2x^2-x=0}}}  Subtract x from both sides. 





{{{x(x+1)(3x-1)=0}}} Factor the left side 



Now set each factor equal to zero:

{{{x=0}}}, {{{x+1=0}}} or {{{3x-1=0}}}


{{{x=0}}}, {{{x=-1}}} or {{{x=1/3}}}   Now solve for x in each case



So our answer is 

 {{{x=0}}}, {{{x=-1}}} or {{{x=1/3}}}



Notice if we graph {{{y=3x^3+2x^2-x}}}  we can see that the roots are {{{x=0}}}, {{{x=-1}}} and {{{x=1/3}}}. So this visually verifies our answer.



{{{ graph(500,500,-5,5,-5,5,0, 3x^3+2x^2-x) }}}