The sum of two numbers is 9 and the sum of their squares is 261. How do you find the numbers?

2 Answers
Apr 28, 2017

Either 15 and -6 or -6 and 15.

Explanation:

Let one number be x, then other's is 9 - x.

As per questions,

#x^2+(9-x)^2 = 261#

#rArr x^2+81-18x+x^2 = 261#

#rArr 2x^2-18x-180 = 0#

#rArr x^2-9x-90 = 0#

#rArr x^2-15x+6x-90 = 0#

#rArr x(x-15)+6(x-15) = 0#

#rArr (x-15)(x+6)= 0#

#rArr x = 15 or -6#

other's number is #(9 - 15) = - 6 or (9+ 6) = 15#
Hence numbers are either 15 and -6 or -6 and 15.

Apr 28, 2017

#-6 " and " 15#

Explanation:

Given: Sum of the two numbers is #9: " "x + y = 9#

Given: Sum of the squares is #261: " "x^2 + y^2 = 261#

Use substitution by solving for one variable in the first equation and substituting this variable into the second equation:

#x = 9 - y#

#(9 - y)^2 + y^2 = 261#

Distribute using #(a - b)^2 = (a^2 - 2ab + b^2)#:

#81 - 18y + y^2 + y^2 = 261#

Simplify: # 2y^2 - 18y +81 - 261 = 0#

#2y^2 - 18y +180 = 0#

Factor a #2: " "2(y^2 - 9y + 90) = 0#

Factor quadratic: #2(y - 15)(y + 6) = 0#

Solve for #y#:

#y - 15 = 0; " " y = 15 " and " y + 6 = 0; " " y = -6#

Solve for #x#:

#" "x = 9 - 15 = -6 " and " x = 9 - -6 = 9 + 6 = 15#

Solution: #-6 " and " 15#