Find two consecutive odd integers such that the difference of their squares is 160.?

Answer 1

(x+2)^2-x^2=160
(x+2+x)(x+2-x)=160
2x+2=80
2x=78
x=39
so the nos are 39 and 41

Answer 2

First, let's call the smaller number x. Then the consecutive odd integer will be x+2.

Our equation would be:
(x+2)^2 - x^2 = 160

Expand the square to get (Alternatively, you may use the identity for differences of squares):
x^2 + 4x + 4 - x^2 = 160

The x^2 terms will cancel out:
4x + 4 = 160
4x = 154
x = 39

So, the smaller number is 39. Then the larger one would be 41.

Now, let's check our answer:
41^2 = 1681
39^2 = 1521
1681 - 1521 = 160

So, our numbers are correct!

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Another approach to this question might be calling the even number between our consecutive odd integers x.

Then the smaller number would be x-1 and the larger one x+1. So, we have:
(x+1)^2 - (x-1)^2 = 160

Here, use the identity a^2 - b^2 = (a-b)(a+b).
((x+1)+(x-1)) * ((x+1)-(x-1)) = 160
(2x) * 2 = 160
4x = 160
x = 40

So:
x-1 = 39
x+1 = 41

Again, we found the same numbers as the answer.

Answer 3

Let the integers be 'x' and 'x + 2'
Turn your word problem into an equation.
The required equation is:

(x + 2)^2 - x^2 = 160
x^2 + 4x + 4 - x^2 = 160
4x + 4 = 160
4(x + 1) = 160
x + 1 = 160/4
x + 1 = 40
x = 39
x + 2 = 41

The numbers are 39 and 41

Check:
41^2 - 39^2
1681 - 1521 = 160

Answer 4

You can approach this question via 2 methods. Trial and error (randomly guessing 2 consecutive odd integers that give you 160) or by algebra, depending on your level. Trial and error for lower grade and algebra for advanced grade.

I'll show you the method by algebra.
Let x be the smaller odd integer and (x+2) be the larger odd integer.
(x+2)²-x² = 160
Expanding x(x+2)
x²+4x+4 - x²=160
4x+4 =160
4x = 160 - 4
4x = 156
x = 39
x+2= 41
Thus the 2 consecutive odd integers are 39 and 41.

Note (a+b)² = a²+2ab+b²

Answer 5

Let the smaller number be x
since it is consecutive number s, the second number will be (x+2)
ex : 1,3,5
The differences are 2

The, we know that
(x+2)^2-x^2=160
x^2+4x+4-x^2=160
4x=156
x=39

So, the smaller number is 39 while the bigger is 41