Question 196582
Since "the image of an object is 4 cm closer to the lens than the object itself", this means that {{{y=x-4}}}



{{{1/F = 1/x + 1/y }}} Start with the given equation.



{{{1/2.1 = 1/x + 1/(x-4)}}} Plug in {{{F=2.1}}} and {{{y=x-4}}}



{{{cross(2.1)x(x-4)(1/cross(2.1)) = 2.1*cross(x)(x-4)(1/cross(x)) + 2.1x*cross((x-4))(1/cross((x-4)))}}} Multiply EVERY term by the LCD {{{2.1x(x-4)}}} to clear out the fractions.



{{{x(x-4)=2.1(x-4)+2.1x(1)}}} Cancel out and simplify



{{{x(x-4)=2.1(x-4)+2.1x}}} Multiply



{{{x^2-4x=2.1x-8.4+2.1x}}} Distribute



{{{10x^2-40x=21x-84+21x}}} Multiply EVERY term by 10 to make every number a whole number.



{{{10x^2-40x-21x+84-21x=0}}} Get all terms to the left side.



{{{10x^2-82x+84=0}}} Combine like terms.



Notice we have a quadratic in the form of {{{Ax^2+Bx+C}}} where {{{A=10}}}, {{{B=-82}}}, and {{{C=84}}}



Let's use the quadratic formula to solve for x



{{{x = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{x = (-(-82) +- sqrt( (-82)^2-4(10)(84) ))/(2(10))}}} Plug in  {{{A=10}}}, {{{B=-82}}}, and {{{C=84}}}



{{{x = (82 +- sqrt( (-82)^2-4(10)(84) ))/(2(10))}}} Negate {{{-82}}} to get {{{82}}}. 



{{{x = (82 +- sqrt( 6724-4(10)(84) ))/(2(10))}}} Square {{{-82}}} to get {{{6724}}}. 



{{{x = (82 +- sqrt( 6724-3360 ))/(2(10))}}} Multiply {{{4(10)(84)}}} to get {{{3360}}}



{{{x = (82 +- sqrt( 3364 ))/(2(10))}}} Subtract {{{3360}}} from {{{6724}}} to get {{{3364}}}



{{{x = (82 +- sqrt( 3364 ))/(20)}}} Multiply {{{2}}} and {{{10}}} to get {{{20}}}. 



{{{x = (82 +- 58)/(20)}}} Take the square root of {{{3364}}} to get {{{58}}}. 



{{{x = (82 + 58)/(20)}}} or {{{x = (82 - 58)/(20)}}} Break up the expression. 



{{{x = (140)/(20)}}} or {{{x =  (24)/(20)}}} Combine like terms. 



{{{x = 7}}} or {{{x = 6/5}}} Simplify. 



So the <i>possible</i> answers are {{{x = 7}}} or {{{x = 6/5}}}  

  

However, since {{{6/5-4=1.2-4=-2.8}}} (which doesn't make much sense), this means we'll ignore  {{{x = 6/5}}}



So the only answer is {{{x = 7}}}



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Answer:


So the object is 7 cm from the lens.