Question 202846
{{{e^(4x^2+12x+9)=1}}} Start with the given equation.



{{{ln(e^(4x^2+12x+9))=ln(1)}}} Take the natural log of both sides



{{{4x^2+12x+9=ln(1)}}} Evaluate the natural log of {{{e^(4x^2+12x+9)}}} to get {{{4x^2+12x+9}}}



{{{4x^2+12x+9=0}}} Evaluate the natural log of 1 to get 0



Notice that the quadratic {{{4x^2+12x+9}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=4}}}, {{{B=12}}}, 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 = (-(12) +- sqrt( (12)^2-4(4)(9) ))/(2(4))}}} Plug in  {{{A=4}}}, {{{B=12}}}, and {{{C=9}}}



{{{x = (-12 +- sqrt( 144-4(4)(9) ))/(2(4))}}} Square {{{12}}} to get {{{144}}}. 



{{{x = (-12 +- sqrt( 144-144 ))/(2(4))}}} Multiply {{{4(4)(9)}}} to get {{{144}}}



{{{x = (-12 +- sqrt( 0 ))/(2(4))}}} Subtract {{{144}}} from {{{144}}} to get {{{0}}}



{{{x = (-12 +- sqrt( 0 ))/(8)}}} Multiply {{{2}}} and {{{4}}} to get {{{8}}}. 



{{{x = (-12 +- 0)/(8)}}} Take the square root of {{{0}}} to get {{{0}}}. 



{{{x = (-12 + 0)/(8)}}} or {{{x = (-12 - 0)/(8)}}} Break up the expression. 



{{{x = (-12)/(8)}}} or {{{x =  (-12)/(8)}}} Combine like terms. 



{{{x = -3/2}}} or {{{x = -3/2}}} Simplify. 



So the only solution is {{{x = -3/2}}}