Question 177030
Take note that the matrix {{{(matrix(3,3,4,x,-2,-x,-3,1,-6,2,3))}}} is of the form 
{{{abs(matrix(3,3,a,b,c,d,e,f,g,h,i))}}} where {{{a=4}}}, {{{b=x}}}, {{{c=-2}}}, {{{d=-x}}}, {{{e=-3}}}, {{{f=1}}}, {{{g=-6}}}, {{{h=2}}}, and {{{i=3}}}



Since {{{abs(matrix(3,3,4,x,-2,-x,-3,1,-6,2,3))=-3}}}, this means that {{{d=-3}}}




{{{abs(matrix(3,3,a,b,c,d,e,f,g,h,i))=a(ei-fh)-b*(di-fg)+c(dh-eg)}}} Start with the general 3x3 <a href="http://en.wikipedia.org/wiki/Determinant#3-by-3_matrices">determinant</a> formula.




{{{abs(matrix(3,3,4,x,-2,-x,-3,1,-6,2,3))=(4)((-3)(3)-(1)(2))-(x)((-x)(3)-(1)(-6))+(-2)((-x)(2)-(-3)(-6))}}} Plug in the given values (see above)



{{{d=(4)((-3)(3)-(1)(2))-(x)((-x)(3)-(1)(-6))+(-2)((-x)(2)-(-3)(-6))}}} Replace the left side with "d" (see above)



{{{d=4(-9-2)-x(-3x+6)-2(-2x-18)}}} Multiply



{{{d=4(-11)-x(-3x+6)-2(-2x-18)}}} Combine like terms. 



{{{d=-44-x(-3x+6)-2(-2x-18)}}} Multiply



{{{d=-44+3x^2-6x+4x+36}}} Distribute



{{{-3=-44+3x^2-6x+4x+36}}} Plug in {{{d=-3}}}



{{{0=-44+3x^2-6x+4x+36+3}}} Add 3 to both sides.



{{{0=3x^2-2x-5}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=3}}}, {{{b=-2}}}, and {{{c=-5}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-2) +- sqrt( (-2)^2-4(3)(-5) ))/(2(3))}}} Plug in  {{{a=3}}}, {{{b=-2}}}, and {{{c=-5}}}



{{{x = (2 +- sqrt( (-2)^2-4(3)(-5) ))/(2(3))}}} Negate {{{-2}}} to get {{{2}}}. 



{{{x = (2 +- sqrt( 4-4(3)(-5) ))/(2(3))}}} Square {{{-2}}} to get {{{4}}}. 



{{{x = (2 +- sqrt( 4--60 ))/(2(3))}}} Multiply {{{4(3)(-5)}}} to get {{{-60}}}



{{{x = (2 +- sqrt( 4+60 ))/(2(3))}}} Rewrite {{{sqrt(4--60)}}} as {{{sqrt(4+60)}}}



{{{x = (2 +- sqrt( 64 ))/(2(3))}}} Add {{{4}}} to {{{60}}} to get {{{64}}}



{{{x = (2 +- sqrt( 64 ))/(6)}}} Multiply {{{2}}} and {{{3}}} to get {{{6}}}. 



{{{x = (2 +- 8)/(6)}}} Take the square root of {{{64}}} to get {{{8}}}. 



{{{x = (2 + 8)/(6)}}} or {{{x = (2 - 8)/(6)}}} Break up the expression. 



{{{x = (10)/(6)}}} or {{{x =  (-6)/(6)}}} Combine like terms. 



{{{x = 5/3}}} or {{{x = -1}}} Simplify. 



So the answers are {{{x = 5/3}}} or {{{x = -1}}}