Question 193113


{{{4x^2-4x=1}}} Start with the given equation.



{{{4x^2-4x-1=0}}} Subtract 1 from both sides.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=4}}}, {{{b=-4}}}, 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 = (-(-4) +- sqrt( (-4)^2-4(4)(-1) ))/(2(4))}}} Plug in  {{{a=4}}}, {{{b=-4}}}, and {{{c=-1}}}



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



{{{x = (4 +- sqrt( 16-4(4)(-1) ))/(2(4))}}} Square {{{-4}}} to get {{{16}}}. 



{{{x = (4 +- sqrt( 16--16 ))/(2(4))}}} Multiply {{{4(4)(-1)}}} to get {{{-16}}}



{{{x = (4 +- sqrt( 16+16 ))/(2(4))}}} Rewrite {{{sqrt(16--16)}}} as {{{sqrt(16+16)}}}



{{{x = (4 +- sqrt( 32 ))/(2(4))}}} Add {{{16}}} to {{{16}}} to get {{{32}}}



{{{x = (4 +- sqrt( 32 ))/(8)}}} Multiply {{{2}}} and {{{4}}} to get {{{8}}}. 



{{{x = (4 +- 4*sqrt(2))/(8)}}} Simplify the square root  



{{{x = (4+4*sqrt(2))/(8)}}} or {{{x = (4-4*sqrt(2))/(8)}}} Break up the expression.  



{{{x = (1+sqrt(2))/(2)}}} or {{{x = (1-sqrt(2))/(2)}}} Reduce



So the answers are {{{x = (1+sqrt(2))/(2)}}} or {{{x = (1-sqrt(2))/(2)}}} 



which approximate to {{{x=1.207}}} or {{{x=-0.207}}}