Find 2 consecutive integers such that the square of the sum of the 2 integers is 11 more than the smaller in?

Question

Find two consecutive integers such that the square of the sum of the two integers is 11 more than ther smaller integer.

Answer 1

your integers are x, x+1 .... so let's set it up as an algebra problem, like this:

"the square of the sum" would be (x+x+1)^2 = (2x+1)^2

"11 more than the smaller" means x+11

So, we have (2x+1)^2 = x + 11

Now, multiply out, and solve for x:

4x^2 + 4x + 1 = x + 11

4x^2 + 3x - 10 = 0

(4x - 5) (x + 2) = 0

So, x = 5/4 or -2

Since we're interested in an integer, we choose x = -2

Our integers are -2 and -1. Now let's plug back in to our original problem to check our results:

The square of the sum is 9 (-2-1 = -3, -3^2 = 9)

9 is 11 more than the smaller (-2) so this works!

{{ FYI please note: The answer of 2 & 3 below is wrong. The solver took 4 + 9 which is the sum of the squares, not the square of the sum! }}

Answer 2

Let the two integers be 'x' and 'x+1'
(x+x+1)^2 = x+11
(2x + 1)^2 = x+11
4x^2 + 1 + 4x = x+11
4x^2 + 4x = x + 10
4x^2 + 3x = 10
4x^2 + 3x -10 = 0
By factorising the quadratic polynomial,
(4x - 5)(x+2) = 0
Either 4x -5 or x+2 =0
Putting 4x-5 = 0
4x = 5
x = 5/4
This possibility can be safely eliminated as we want an integer
Put x+2 = 0
x = -2 which is an integer.
Therefore the integers are -2 and -1.
Let's check the solution:
LHS=(-2-1)^2
= -3^2
= 9
RHS=-2 + 11
= 9
LHS = RHS
Our solution is correct

Answer 3

Let the integers be x, x+1;
Therefore, (x+x+1)^2 - x =11
Or, 4*x^2 + 4*x +1 - x = 11
Or, 4*x^2 + 3*x -10 = 0
Or, (4*x^2 + 8*x) - 5*x -10 =0
Or, 4*x ( x + 2) - 5( x+2) =0
Or, x= 5/4 or x =-2

The two integers are -2 and -1

Answer 4

-2, -1 or 5/4, 9/4
(x+x+1)^2=11+x

Answer 5

Its 2 & 3

as 4 + 9 = 13, which is 11 more than 2 :)

Answer 6

let the number be x
so the consecutive no is x+1
adding both we get x+(x+1) = 2x+1
the equation is :
(2x+1) ^2 - x =11
i.e., 4x^2 +3x-10 = 0
solving we get x = 1.25, -2
But since we need an integer the answer is -2.

Cross check :
-2-1 =-3
sqaring we get 9
9-(-2) = 11

Answer 7

-2 and -1 did i beat the other guys?


post-edit: no, no i didn't