Question 192823
Please try to not use all caps (it is hard to read).



"The sum of an integer and its square (that integer multiplied times itself) is 132" translates to {{{x+x^2=132}}}



{{{x+x^2=132}}} Start with the given equation.



{{{x+x^2-132=0}}} Subtract 132 from both sides.



{{{x^2+x-132=0}}} Rearrange the terms.




Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=1}}}, and {{{c=-132}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(1) +- sqrt( (1)^2-4(1)(-132) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=1}}}, and {{{c=-132}}}



{{{x = (-1 +- sqrt( 1-4(1)(-132) ))/(2(1))}}} Square {{{1}}} to get {{{1}}}. 



{{{x = (-1 +- sqrt( 1--528 ))/(2(1))}}} Multiply {{{4(1)(-132)}}} to get {{{-528}}}



{{{x = (-1 +- sqrt( 1+528 ))/(2(1))}}} Rewrite {{{sqrt(1--528)}}} as {{{sqrt(1+528)}}}



{{{x = (-1 +- sqrt( 529 ))/(2(1))}}} Add {{{1}}} to {{{528}}} to get {{{529}}}



{{{x = (-1 +- sqrt( 529 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (-1 +- 23)/(2)}}} Take the square root of {{{529}}} to get {{{23}}}. 



{{{x = (-1 + 23)/(2)}}} or {{{x = (-1 - 23)/(2)}}} Break up the expression. 



{{{x = (22)/(2)}}} or {{{x =  (-24)/(2)}}} Combine like terms. 



{{{x = 11}}} or {{{x = -12}}} Simplify. 



So the answers are {{{x = 11}}} or {{{x = -12}}} 



So the number is either 11 or -12.



Note: if you only care about positive numbers, then the answer is 11.