The sum of the squares of two consecutive numbers is 1105. What are the two numbers?

Answer 1

x² + (x+1)² = 1105
2x² + 2x + 1 = 1105
x² + x - 552 = 0
Solve the quadratic and get 23 and 24 as the numbers.

BTW, -24 and -23 also work ☺


Doug

Answer 2

The numbers are x and x + 1. Their squares are x^2 and (x + 1)^2 = x^2 + 2*x + 1. Their sum is 2*x^2 + 2*x + 1. This equals 1105. Solve for x using the quadratic formula.

Answer 3

let the numbers be x, x+1 then x^2+(x+1)^2=1105
x^2+x^2+2x+1=1105
2x^2+2x+1=1105
x^2+x-552=0
(x+24)(x-23)=0
x=23,x=-24
the possitive number is x=23
x=23, x+1=24
and negative number is x=-24
x=-24, x+1=-23

Answer 4

x^2 +(x +1)^2 =1105
x^2 +x^2 +2x +1 =1105
2*x^2 +2x +1 =1105

2*x^2 +2x -1104 =0
x^2 +x -552 =0

now solve for x using quadratic equation

Or you can guess-timate
approximate x by doing the following
x approx squareroot(1105/2)
23.505

test 23^2 +24^2
529 + 576 =1105

Answer 5

x²+(x+1)²=1105
x²+x²+2x+1=1105
2x²+2x+1=1105
2x²+2x+1-1105=0
2x²+2x-1104=0
2(x²+x-552)=0
x²+x-552=0
From here we must find two numbers such that their product is -552 and their sum is 1. If we apply the quadratic equation [-b ± √(b²-4ac)]/2a, the roots would be 23 and 24.
So the two numbers are 23 and 24.
Let's check:
23²+24²=1105
529+576=1105
1105=1105
So, the answers are correct. 23 and 24 really are the answers.

Answer 6

x^2 + (x+1)^2 = 1105

x^2 + x^2 + 2x + 1 = 1105

2x^2 + 2x - 1104 = 0

x^2 + x - 552 = 0

Then you solve this equation

Another form:

1105 / 2 = 552,5

So, we only have to find a square that is bigger than 552,5 and the one who is less than 552,5

Ana

Answer 7

1105 = n^2 + (n+1)^2 = 2n^2 + 2n + 1
2n^2 + 2n - 1104 = 0

n = (-2 + sqrt(4 + 4*2*1104)) / 4 = 23

also

n = (-2 - sqrt(4 + 4*2*1104)) / 4 = -24

so the pair 23/24 and the pair -24/-23 satisfy the requirements.

Answer 8

x = 1st no.
x+1 = next no.

x^2, (x+1)^2

x^2 + (x+1)^2 = 1105
x^2 + x^2 + 2x + 1 = 1105
2x^2 + 2x -1104 = 0
(2x + 48) (x - 23) = 0
(2x + 48) = 0 or (x - 23)=0

(2x + 48) = 0
x = -24
x+1 = -24 + 1
x+1= -23
(-24, -23)

(x - 23)=0
x = 23
x+1 = 23+1
x+1 = 24
(23, 24)

Answer 9

(x)^2 + (x+1)^2 = 1105
(x)^2 + (x+1)(x+1) = 1105
(x)^2 + (x)^2 + 2x +1 = 1105
2(x)^2 + 2x + 1 = 1105
2(x)^2 + 2x - 1104 = 0
Quadratic Formula
(-2) +/- sq. root(2^2 - 4*2*(-1104))/(2*2)
(-2) +/- sq root(4 - (-8832))/(4)
(-2) +/- sq root(8836)/(4)
(-2) +/- 94/(4)
92/4 or (-96)/4
23 -24

So that means either (23,24) or (-24,-23)

Answer 10

23 and 24 or -24 and -23.