What is the sum of two consecutive even numbers if the difference of whose squares is 92

The difference of the squares of two consecutive even natural numbers is 68. Find the two numbers.

Let the two consecutive even natural numbers be 2x and 2x + 2.As per the given condition,

(2x + 2)2 - (2x)2 = 68

⇒ 4x2 + 8x + 4 - 4x2 = 68⇒ 8x + 4 = 68

⇒ 8x = 64

⇒ x = `(64)/(8)`⇒ x = 82x = 2(8) = 16 and 2x + 2 = 2(8) + 2 = 16 + 2 = 18

Hence, the numbers are 16 and 18.

Concept: Simple Linear Equations in One Variable

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Page 2

The sum of the squares of two consecutive odd natural numbers is 650. Find the two numbers.

Let the two consecutive od number be 2n + 1 and 2n + 3, where n is a natural number.
Given, (2n + 3)2 + (2n + 1)2 = 650
⇒ (4n2 + 9 + 12n) + (4n2 + 1 + 4n) = 650
⇒ 8n2 + 10 + 16n = 650
⇒ 8n2 + 16n - 640 = 0
⇒ n2 + 2n - 80 = 0
⇒ n2 + 10n - 8n - 80 = 0⇒ n(n + 10) - 8(n + 10) = 0⇒ (n - 8) (n + 10) = 0Thus, n = 8 as n = -10 is not possible.

Thus, the two consecutive od natural numbers are 17 and 19.

Concept: Simple Linear Equations in One Variable

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Discussion :: Problems on Numbers - General Questions (Q.No.15)

15.

What is the sum of two consecutive even numbers, the difference of whose squares is 84?

[A].	34
[B].	38
[C].	42
[D].	46

Answer: Option C

Explanation:

Let the numbers be x and x + 2.

Then, (x + 2)2 - x2 = 84

4x + 4 = 84

4x = 80

x = 20.

The required sum = x + (x + 2) = 2x + 2 = 42.

Sunil said: (Jul 7, 2011)
The questions is for sum of two consecutive "even numbers" So in your answer if x=1 then x+2=3 Both are odd numbers. Therefore i think the solution may be as follows:- Let the numbers be "2x" and "2x + 2". Then, (2x + 2)^2 - 2x^2 = 84 8x+4=84 x=10. The required sum = 2x + (2x + 2) = 42.

Pratheep Msajce It said: (Jul 13, 2011)
The difference of the squares of the number is 84 So (a^2)-(b^2)=84 then (a+b)(a-b)=84 (a-b)=2[it is difference between consecutive even no] then (a+b)*(2)=84 A+B=42

Arafath said: (Jul 6, 2012)
How does x square become 4x. ?

Nivedha said: (Mar 17, 2013)
But, if we take x^2-(x+)^2 = 84, The answer changes to 46. Can anyone explain why they take (x+2)^2-x^2 = 84.

Kranti said: (Jul 13, 2014)
@Nivedha ji, (x+2) is greater than x... and as it is for subtraction, he has taken larger number first, for getting a positive number.

Jack said: (May 14, 2015)
I think it should be. (x^2) - (x+1)^2. (x+x+1)(x-x+1) = 84 //Using a^2 -b^2 formula. (2x+1)(1) = 84. 2x+1= 84. 2x = 83. x = 83/2.

Pooja said: (Nov 6, 2016)
if we took like that (x)^2-(x+2)^2 = 84, answer is changed i.e.46. Is it correct?

Sankarilakshmi said: (Nov 17, 2016)
Hi @Pooja. If we took like that (x)^2 - (x+2)^2 = 84. Then, x^2-(x^2+4x+4) = 84, x^2-x^2-4x-4 = 84, -4x = 88, x = -22. The required sum = x + (x + 2). = -22 + (-22 + 2), =-22 - 20 = -42 (same answer but negative integer so we taken the large number first).

Pranshu Yadav said: (May 10, 2017)
In this case; Consecutive even no x, x+2. According to the question. (x)^2 - (x+2)^2 = 84. (x+x+2)(x-x+2) = 84. (2x + 2)2 = 84. 4x + 4 = 84. 4x =84-4. 4x=80. x=80/4. x=20. Then; = x +x+2. = 20+20+2. = 42. This is the correct answer = 42.

Sundar Ds said: (May 23, 2017)
(x+2)2/x2 = 84, x2 + 4 + 4x/x2 = 84, 4x+4 = 84, 4x = 84 - 4, 4x = 80, x = 20. The required sum=x + (x+2) 2x + 2, 2 * 20+2 = 42.

Nang said: (Jul 1, 2017)
Let two consecutive no. Be x, x+2, Then ,x-x+2=84, = 84/2, = 42.

Soumya said: (Mar 28, 2018)
Let two consecutive even numbers be 2x,2(x+1)[taking x is not valid] so (2x+2)^2-4x^=84. or, 8x-4=84. or, x=11. So two numbers are 211,212= 22,24. Sum = 22+24= 46.

Tushar said: (Jul 27, 2019)
Let two consecutive number Be x, x+2. Then, x^2 - (x+2)^2 = 84; So put (a+b)^2 formulae, Then x^2-(x^2+2x2+2^2)=84; cut x^2,remaining is 4x+4=84; x=20; And put this in equation 2x+2=2*20+2=42.

Pavinesh said: (Aug 4, 2019)
TAKE 2x ,2x+2 be the two consecutive numbers: Then as per given condition, (2x+2)^2-(2x)^2=84. (4x^2+4+8x)-(4x^2)=84 8x=80 x=10; If you substitute this to 2x+2x+2. 20+20+2=42. I hope you get a clear solution.

Anshu Saini said: (Jan 22, 2021)
Two consecutive even numbers can only be 2x and 2x+2. While x and x+2 are example of two consecutive odd numbers.

Punam Paswan said: (Apr 8, 2021)
The consecutive no Be x, x+2, Then, x-x+2=84, = 84/2, = 42.

Subas said: (Jul 10, 2021)
Let first even number is (x) and the second even number is (x+2). According to question; (x+2)^2-x^2 = 84. (x+2-x)(x+2+x) = 84, 2(2x+2) = 42, 2x+2 = 42, 2x = 40, x = 20. Next no.= 20 + 2 = 22, Sum = 20 + 22 + 42.