Question 151436
"2 positive numbers that differ by 2" translates to {{{x-y=2}}} and "have a product of 20" translates to {{{x*y=20}}}



{{{x-y=2}}} Start with the first equation.



{{{x=2+y}}} Add {{{y}}} to both sides.



{{{x-2=y}}} Subtract 2 from both sides.



So after isolating "y", we get {{{y=x-2}}}


{{{x*y=20}}} Move onto the second equation



{{{x*(x-2)=20}}} Plug in {{{y=x-2}}}



{{{x^2-2x=20}}} Distribute.



{{{x^2-2x-20=0}}} Subtract 20 from both sides.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-2}}}, and {{{c=-20}}}



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)(-20) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-2}}}, and {{{c=-20}}}



{{{x = (2 +- sqrt( (-2)^2-4(1)(-20) ))/(2(1))}}} Negate {{{-2}}} to get {{{2}}}. 



{{{x = (2 +- sqrt( 4-4(1)(-20) ))/(2(1))}}} Square {{{-2}}} to get {{{4}}}. 



{{{x = (2 +- sqrt( 4--80 ))/(2(1))}}} Multiply {{{4(1)(-20)}}} to get {{{-80}}}



{{{x = (2 +- sqrt( 4+80 ))/(2(1))}}} Rewrite {{{sqrt(4--80)}}} as {{{sqrt(4+80)}}}



{{{x = (2 +- sqrt( 84 ))/(2(1))}}} Add {{{4}}} to {{{80}}} to get {{{84}}}



{{{x = (2 +- sqrt( 84 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (2 +- 2*sqrt(21))/(2)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{x = (2)/(2) +- (2*sqrt(21))/(2)}}} Break up the fraction.  



{{{x = 1 +- sqrt(21)}}} Reduce.  



{{{x = 1+sqrt(21)}}} or {{{x = 1-sqrt(21)}}} Break up the expression.  



So the values of x are {{{x = 1+sqrt(21)}}} or {{{x = 1-sqrt(21)}}} 



which approximate to {{{x=5.583}}} or {{{x=-3.583}}}




{{{y=x-2}}} Go back to the first isolated equation



{{{y=1+sqrt(21)-2}}} Plug in {{{x = 1+sqrt(21)}}}



{{{y=-1+sqrt(21)}}} Combine like terms.



So the first pair of numbers is {{{x = 1+sqrt(21)}}} and {{{y=-1+sqrt(21)}}}


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{{{y=x-2}}} Go back to the first isolated equation



{{{y=1-sqrt(21)-2}}} Plug in {{{x = 1-sqrt(21)}}}



{{{y=-1-sqrt(21)}}} Combine like terms.



So the next pair of numbers is {{{x = 1-sqrt(21)}}} and {{{y=-1-sqrt(21)}}}