SOLUTION: How would you find 2 positive numbers that differ by 2 and have a product of 20? I dont know how to set up the equation in this instance.

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: How would you find 2 positive numbers that differ by 2 and have a product of 20? I dont know how to set up the equation in this instance.      Log On


   

Question 151436: How would you find 2 positive numbers that differ by 2 and have a product of 20? I dont know how to set up the equation in this instance.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"2 positive numbers that differ by 2" translates to x-y=2 and "have a product of 20" translates to x%2Ay=20


x-y=2 Start with the first equation.


x=2%2By Add y to both sides.


x-2=y Subtract 2 from both sides.


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

x%2Ay=20 Move onto the second equation


x%2A%28x-2%29=20 Plug in y=x-2


x%5E2-2x=20 Distribute.


x%5E2-2x-20=0 Subtract 20 from both sides.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=-2, and c=-20


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-20%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-2, and c=-20


x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-20%29+%29%29%2F%282%281%29%29 Negate -2 to get 2.


x+=+%282+%2B-+sqrt%28+4-4%281%29%28-20%29+%29%29%2F%282%281%29%29 Square -2 to get 4.


x+=+%282+%2B-+sqrt%28+4--80+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-20%29 to get -80


x+=+%282+%2B-+sqrt%28+4%2B80+%29%29%2F%282%281%29%29 Rewrite sqrt%284--80%29 as sqrt%284%2B80%29


x+=+%282+%2B-+sqrt%28+84+%29%29%2F%282%281%29%29 Add 4 to 80 to get 84


x+=+%282+%2B-+sqrt%28+84+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%282+%2B-+2%2Asqrt%2821%29%29%2F%282%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%282%29%2F%282%29+%2B-+%282%2Asqrt%2821%29%29%2F%282%29 Break up the fraction.


x+=+1+%2B-+sqrt%2821%29 Reduce.


x+=+1%2Bsqrt%2821%29 or x+=+1-sqrt%2821%29 Break up the expression.


So the values of x are x+=+1%2Bsqrt%2821%29 or x+=+1-sqrt%2821%29


which approximate to x=5.583 or x=-3.583



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


y=1%2Bsqrt%2821%29-2 Plug in x+=+1%2Bsqrt%2821%29


y=-1%2Bsqrt%2821%29 Combine like terms.


So the first pair of numbers is x+=+1%2Bsqrt%2821%29 and y=-1%2Bsqrt%2821%29

-------------------------


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


y=1-sqrt%2821%29-2 Plug in x+=+1-sqrt%2821%29


y=-1-sqrt%2821%29 Combine like terms.


So the next pair of numbers is x+=+1-sqrt%2821%29 and y=-1-sqrt%2821%29