Question 175637
{{{3x^2-1=5x^2-3x-5}}} Start with the given equation.



{{{0=5x^2-3x-5-3x^2+1}}} Subtract {{{3x^2}}} from both sides. Add 1 to both sides.



{{{0=2x^2-3x-4}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=2}}}, {{{b=-3}}}, and {{{c=-4}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-3) +- sqrt( (-3)^2-4(2)(-4) ))/(2(2))}}} Plug in  {{{a=2}}}, {{{b=-3}}}, and {{{c=-4}}}



{{{x = (3 +- sqrt( (-3)^2-4(2)(-4) ))/(2(2))}}} Negate {{{-3}}} to get {{{3}}}. 



{{{x = (3 +- sqrt( 9-4(2)(-4) ))/(2(2))}}} Square {{{-3}}} to get {{{9}}}. 



{{{x = (3 +- sqrt( 9--32 ))/(2(2))}}} Multiply {{{4(2)(-4)}}} to get {{{-32}}}



{{{x = (3 +- sqrt( 9+32 ))/(2(2))}}} Rewrite {{{sqrt(9--32)}}} as {{{sqrt(9+32)}}}



{{{x = (3 +- sqrt( 41 ))/(2(2))}}} Add {{{9}}} to {{{32}}} to get {{{41}}}



{{{x = (3 +- sqrt( 41 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{x = (3+sqrt(41))/(4)}}} or {{{x = (3-sqrt(41))/(4)}}} Break up the expression.  



So the answers are {{{x = (3+sqrt(41))/(4)}}} or {{{x = (3-sqrt(41))/(4)}}} 



which approximate to {{{x=2.351}}} or {{{x=-0.851}}}