Question

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

Answer 1

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.

Answer 2

`-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`