The difference between two positive numbers is 4 and the difference between their cubes is 316. Find : (i) their product (ii) the sum of their squares.

Let two positive numbers be x and y with x y. Given, The difference between two positive numbers is 4 and the difference between their cubes is 316. x - y = 4 and x 3 - y 3 = 316. (i) Given, ⇒ x - y = 4 Cubing both sides we get : ⇒ (x - y) 3 = 4 3 ⇒ x 3 - y 3 - 3xy(x - y) = 64 ⇒ 316 - 3xy × 4 = 64 ⇒ 316 - 12xy = 64 ⇒ 12xy = 316 - 64 ⇒ 12xy = 252 ⇒ xy = = 21. Hence, product of numbers = 21. (ii) Given, x - y = 4 Squaring both sides we get : ⇒ (x - y) 2 = 4 2 ⇒ x 2 + y 2 - 2xy = 16 ⇒ x 2 + y 2 - 2 × 21 = 16 ⇒ x 2 + y 2 - 42 = 16 ⇒ x 2 + y 2 = 16 + 42 ⇒ x 2 + y 2 = 58. Hence, sum of squares = 58.

In each of the following, find the value of 'a' :
(i) 4x² + ax + 9 = (2x + 3)²
(ii) 4x² + ax + 9 = (2x - 3)²
(iii) 9x² + (7a - 5)x + 25 = (3x + 5)²

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Mathematics

The difference between two positive numbers is 4 and the difference between their cubes is 316. Find :
(i) their product
(ii) the sum of their squares.

Expansions

Related Questions

In each of the following, find the value of 'a' :
(i) 4x² + ax + 9 = (2x + 3)²
(ii) 4x² + ax + 9 = (2x - 3)²
(iii) 9x² + (7a - 5)x + 25 = (3x + 5)²

Mathematics

The difference between two positive numbers is 4 and the difference between their cubes is 316. Find :(i) their product(ii) the sum of their squares.

Expansions

Related Questions

In each of the following, find the value of 'a' :(i) 4x2 + ax + 9 = (2x + 3)2(ii) 4x2 + ax + 9 = (2x - 3)2(iii) 9x2 + (7a - 5)x + 25 = (3x + 5)2

The difference between two positive numbers is 4 and the difference between their cubes is 316. Find :
(i) their product
(ii) the sum of their squares.

In each of the following, find the value of 'a' :
(i) 4x² + ax + 9 = (2x + 3)²
(ii) 4x² + ax + 9 = (2x - 3)²
(iii) 9x² + (7a - 5)x + 25 = (3x + 5)²