Question 189584
First, we need to factor {{{8x^3+27}}}



{{{8x^3+27}}} Start with the left side of the equation.



{{{(2x)^3+(3)^3}}} Rewrite {{{8x^3}}} as {{{(2x)^3}}}. Rewrite {{{27}}} as {{{(3)^3}}}.



{{{(2x+3)((2x)^2-(2x)(3)+(3)^2)}}} Now factor by using the sum of cubes formula. Remember the <a href="http://www.purplemath.com/modules/specfact2.htm">sum of cubes formula</a> is {{{A^3+B^3=(A+B)(A^2-AB+B^2)}}}



{{{(2x+3)(4x^2-6x+9)}}} Multiply



So {{{8x^3+27}}} factors to {{{(2x+3)(4x^2-6x+9)}}}.



In other words, {{{8x^3+27=(2x+3)(4x^2-6x+9)}}}



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So the equation {{{8x^3+27=0}}} then becomes {{{(2x+3)(4x^2-6x+9)=0}}}



{{{(2x+3)(4x^2-6x+9)=0}}} Start with the factored equation.



{{{2x+3=0}}} or {{{4x^2-6x+9=0}}} Set each factor equal to zero



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Let's solve the first equation {{{2x+3=0}}}



{{{2x+3=0}}} Start with the first equation.



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



{{{x=-3/2}}} Divide both sides by 2.



So the first answer is {{{x=-3/2}}}


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Now let's solve the second equation {{{4x^2-6x+9=0}}}



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=4}}}, {{{b=-6}}}, and {{{c=9}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-6) +- sqrt( (-6)^2-4(4)(9) ))/(2(4))}}} Plug in  {{{a=4}}}, {{{b=-6}}}, and {{{c=9}}}



{{{x = (6 +- sqrt( (-6)^2-4(4)(9) ))/(2(4))}}} Negate {{{-6}}} to get {{{6}}}. 



{{{x = (6 +- sqrt( 36-4(4)(9) ))/(2(4))}}} Square {{{-6}}} to get {{{36}}}. 



{{{x = (6 +- sqrt( 36-144 ))/(2(4))}}} Multiply {{{4(4)(9)}}} to get {{{144}}}



{{{x = (6 +- sqrt( -108 ))/(2(4))}}} Subtract {{{144}}} from {{{36}}} to get {{{-108}}}



{{{x = (6 +- sqrt( -108 ))/(8)}}} Multiply {{{2}}} and {{{4}}} to get {{{8}}}. 



{{{x = (6 +- 6i*sqrt(3))/(8)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{x = (6+6i*sqrt(3))/(8)}}} or {{{x = (6-6i*sqrt(3))/(8)}}} Break up the expression.  



{{{x = (3+3i*sqrt(3))/(4)}}} or {{{x = (3-3i*sqrt(3))/(4)}}} Reduce



So the other two answers are {{{x = (3+3i*sqrt(3))/(4)}}} or {{{x = (3-3i*sqrt(3))/(4)}}} 



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Answer:



So the three solutions are {{{x=-3/2}}}, {{{x = (3+3i*sqrt(3))/(4)}}}, or {{{x = (3-3i*sqrt(3))/(4)}}}