Question 196864
First, let's complete the square:



{{{x^2+(3/5)x-1}}} Start with the left side of the given equation.



Take half of the {{{x}}} coefficient {{{3/5}}} to get {{{3/10}}}. In other words, {{{(1/2)(3/5)=3/10}}}.



Now square {{{3/10}}} to get {{{9/100}}}. In other words, {{{(3/10)^2=(3/10)(3/10)=9/100}}}



{{{x^2+(3/5)x+highlight(9/100-9/100)-1}}} Now add <font size=4><b>and</b></font> subtract {{{9/100}}}. Make sure to place this after the "x" term. Notice how {{{9/100-9/100=0}}}. So the expression is not changed.



{{{(x^2+(3/5)x+9/100)-9/100-1}}} Group the first three terms.



{{{(x+3/10)^2-9/100-1}}} Factor {{{x^2+(3/5)x+9/100}}} to get {{{(x+3/10)^2}}}.



{{{(x+3/10)^2-109/100}}} Combine like terms.



So after completing the square, {{{x^2+3/5x-1}}} transforms to {{{(x+3/10)^2-109/100}}}. So {{{x^2+3/5x-1=(x+3/10)^2-109/100}}}.



So {{{x^2+(3/5)x-1=0}}} is equivalent to {{{(x+3/10)^2-109/100=0}}}.



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Now let's solve {{{(x+3/10)^2-109/100=0}}}



{{{(x+3/10)^2-109/100=0}}} Start with the given equation.



{{{(x+3/10)^2=0+109/100}}}Add {{{109/100}}} to both sides.



{{{(x+3/10)^2=109/100}}} Combine like terms.



{{{x+3/10=0+-sqrt(109/100)}}} Take the square root of both sides.



{{{x+3/10=sqrt(109/100)}}} or {{{x+3/10=-sqrt(109/100)}}} Break up the "plus/minus" to form two equations.



{{{x+3/10=sqrt(109)/10}}} or {{{x+3/10=-sqrt(109)/10}}}  Simplify the square root.



{{{x=-3/10+sqrt(109)/10}}} or {{{x=-3/10-sqrt(109)/10}}} Subtract {{{3/10}}} from both sides.



{{{x=(-3+sqrt(109))/(10)}}} or {{{x=(-3-sqrt(109))/(10)}}} Combine the fractions.



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Answer:



So the solutions are {{{x=(-3+sqrt(109))/(10)}}} or {{{x=(-3-sqrt(109))/(10)}}}.