Question 335961
"a student opens a mathematics book to two facing pages. The product of the page numbers is 506" means that {{{x(x+1)=506}}}



{{{x(x+1)=506}}} Start with the given equation.



{{{x^2+x=506}}} Distribute



{{{x^2+x-506=0}}} Subtract 506 from both sides.



Notice that the quadratic {{{x^2+x-506}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=1}}}, {{{B=1}}}, and {{{C=-506}}}



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)(-506) ))/(2(1))}}} Plug in  {{{A=1}}}, {{{B=1}}}, and {{{C=-506}}}



{{{x = (-1 +- sqrt( 1-4(1)(-506) ))/(2(1))}}} Square {{{1}}} to get {{{1}}}. 



{{{x = (-1 +- sqrt( 1--2024 ))/(2(1))}}} Multiply {{{4(1)(-506)}}} to get {{{-2024}}}



{{{x = (-1 +- sqrt( 1+2024 ))/(2(1))}}} Rewrite {{{sqrt(1--2024)}}} as {{{sqrt(1+2024)}}}



{{{x = (-1 +- sqrt( 2025 ))/(2(1))}}} Add {{{1}}} to {{{2024}}} to get {{{2025}}}



{{{x = (-1 +- sqrt( 2025 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (-1 +- 45)/(2)}}} Take the square root of {{{2025}}} to get {{{45}}}. 



{{{x = (-1 + 45)/(2)}}} or {{{x = (-1 - 45)/(2)}}} Break up the expression. 



{{{x = (44)/(2)}}} or {{{x =  (-46)/(2)}}} Combine like terms. 



{{{x = 22}}} or {{{x = -23}}} Simplify. 



So the possible solutions are {{{x = 22}}} or {{{x = -23}}} 

  

Ignore the negative answer as negative page numbers don't make sense.



So the pages are 22 and 23



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