sum and product which both equal to 26

Question

Find a pair of numbers whose sum and product are both equal to 26

(Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list.)

Answer 1

Let x = one of the numbers and y = the other number.

x·y=26 AND x+y=26

solve second equation and get y = 26 - x, then subsitute that into the first equation.

x·(26 - x) = 26

26x - x² = 26

0 = x² - 26x + 26

Need quadratic formula to solve this:

x =

26 ± √[ 676 - (4)(1)(26) ]

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

2(1)

= 26 ± √572

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

2

= 26 ± 2√143

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

2

=13±√143

Answers are: 13 - √143, 13 + √143

Check: (13 - √143)(13 + √143) = 169 -143 = 26 (yay!)

13 - √143 + 13 + √143 = 26 (yay again! -- we got it right!!!! :-) )

Answer 2

Write x(26 − x) = 26 or 26 − 26x + x² = 0.

Then x = {-(-26) ± √[(-26)² − 4(1)26]}/2(1) gives x = {13 + √143}

or x = {13 − √143}.

The two possible solutions are then (x, (26 − x)) equal to (13 + √143,13 − √143) or equal to

(13 − √143,13 + √143).

Answer 3

x*y = 26

x + y = 26

from the 1st equation, x = 26/y

plug this into the 2nd equation to get:

26/y + y = 26

26 + y² =26y

y² - 26y +26 = 0

Does not appear to be factorable, so use the quadratic formula

Answer 4

xy = 26

x + y = 26

y = 26 - x

x(26 - x) = 26

26x - x² = 26

0 = x² - 26x + 26

0 = x² - 26x + 13² + 26 - 13²

0 = (x - 13)² + 26 - 169

0 = (x - 13)² - 143

143 = (x - 13)^2±

±√143 = x - 13

13 ± √143 = x