The integer value of x for which x 2 + 19x + 92 is square o…

Maths Quadratic Equations Solution of Quadratic Equations and Their Roots Single Correct MCQ

Published by: CGP EDU Academic Team

Published on: August 13, 2026

The integer value of x for which x 2 + 19x + 92 is square of an integer are

– 8 & – 11

– 8 & 11

8 & 11

± ± 8, ± ± 8

Text Solution

Verified by Experts

The correct answer is:

Let x 2 + 19x + 92 = m 2 , m I

x 2 + 19x + 92 – m 2 = 0

x =

∴ x I

So, –19 ± ± must be even integer

∴ must be odd integer

Let = 2 k + 1; k I

∴ 4m 2 – (2k + 1) 2 = 7

(2m – 2k – 1) (2m + 2k + 1) = 7

∴ 7 is prime

So, possibilities are

case

(i) 2m – 2k–1=1 & 2m + 2k + 1 = 7 then m = 2

(ii) 2m–2k–1 = 7 & 2m + 2k + 1 = 1 then m = 2

at m = 2 ⇒ x = –8, –11

(iii) 2m – 2k –1 = – 1 & 2m + 2k + 1 = –7

then m = – 2

(iv) 2m – 2k – 1= –7 & 2m + 2k + 1= –1 then m = –2

at m = – 2

x = – 8, – 11

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