Book: RS Aggarwal - Mathematics
Chapter: 10. Quadratic Equations
Subject: Maths - Class 10th
Q. No. 20 of Exercise 10E
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Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.
Let the larger and the smaller parts be x and y respectively.
According to the question
x + y = 16 – – – – – (1)
2x2 = y2 + 164 – – – (2)
From (1) x = 16 – y – – – (3)
From (2) and (3) we get
2(16 – y)2 = y2 + 164
2(256 – 32y + y2) = y2 + 164 using (a + b)2 = a2 + 2ab + b2
512 – 64y + 2y2 = y2 + 164
y2 – 64y + 348 = 0
Using the splitting middle term – the middle term of the general equation 
is divided in two such values that:
Product = a.c
For the given equation a = 1 b = – 64 c = 348
= 1. 348 = 348
And either of their sum or difference = b
= – 64
Thus the two terms are – 58 and – 6
Sum = – 58 – 6 = – 64
Product = – 58. – 6 = 348
y2 – 64y + 348 = 0
y2 – 58y – 6y + 348 = 0
y(y – 58) – 6(y – 58) = 0
(y – 58) (y – 6) = 0
(y – 58) = 0 or (y – 6) = 0
y = 6 (y < 16)
putting the value of y in (3), we get
x = 16 – 6
= 10
Hence the two natural numbers are 6 and 10.
minutes. If one pipe takes 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.
