Question 297338

{{{f(x)=x^2-2x+1}}} Start with the given function.



{{{0=x^2-2x+1}}} Plug in {{{f(x)=0}}}



{{{x^2-2x+1=0}}} Rearrange the equation.



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



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



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



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



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



{{{x = (2 +- sqrt( 0 ))/(2(1))}}} Subtract {{{4}}} from {{{4}}} to get {{{0}}}



{{{x = (2 +- sqrt( 0 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (2 +- 0)/(2)}}} Take the square root of {{{0}}} to get {{{0}}}. 



{{{x = (2 + 0)/(2)}}} or {{{x = (2 - 0)/(2)}}} Break up the expression. 



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



{{{x = 1}}} or {{{x = 1}}} Simplify. 



So the only solution is {{{x = 1}}}