NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 Question 4

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 4

Question:

Find two consecutive positive integers, sum of whose squares is 365.

Given:

Two consecutive positive integers.

Sum of their squares = 365.

To Find:

The two consecutive positive integers.

Formula:

WKT, if the first integer is x, the next consecutive integer is (x + 1).
WKT, the algebraic identity (a + b)2 = a2 + 2ab + b2.

WKT, For a quadratic equation ax² + bx + c = 0, we can find two numbers m and n such that:

                        ax² + bx + c = ax² + mx + nx + c

where

  • m × n = a × c
  • m + n = b

Cases:

  • If (a × c > 0) and b < 0, both m and n are negative.
  • If (a × c > 0) and b > 0, both m and n are positive.
  • If (a × c < 0), the numbers m and n have opposite signs.

Solution:

Let the first positive integer be 'x'.

The next consecutive positive integer will be (x + 1).
According to the given condition, the sum of their squares is 365:

⇒ x2 + (x + 1)2 = 365
Expand (x + 1)2:

⇒ x2 + (x2 + 2x + 1) = 365
Combine like terms:

⇒ 2x2 + 2x + 1 = 365
Rearrange the equation into the standard quadratic form ax2 + bx + c = 0:

⇒ 2x2 + 2x + 1 - 365 = 0

⇒ 2x2 + 2x - 364 = 0
Divide the entire equation by 2 to simplify:

⇒ x2 + x - 182 = 0
Now, we solve this quadratic equation by factorization.

Here, a = 1, b = 1, c = -182.
Product (a × c) = 1 × (-182) = -182.

Sum (b) = 1.
We need to find two numbers whose product is -182 and sum is 1. Take the factors of the product without sign.

WKT, 182 = 2 x 7 x 13

The two numbers we want are m = 14 and n = 13.

Since product is -182 and 182 < 0 both numbers have opposite signs.

Case 1: Let m = -14 and n = 13; product = -182 but sum = -1, therefore the two numbers can't be -14 and 13.

Case 2: Let m = 14 and n = -13; product = -182 and sum = 1, therefore the two numbers are 14 and -13.

Rewrite the middle term (x) using these numbers:

⇒ x2 + 14x - 13x - 182 = 0
Factor by grouping:

⇒ x(x + 14) - 13(x + 14) = 0
Factor out the common binomial (x + 14):

⇒ (x - 13)(x + 14) = 0
Set each factor equal to zero to find the values of x:

Case 1: x - 13 = 0

⇒ x = 13
Case 2: x + 14 = 0

⇒ x = -14
Since we are looking for positive integers, we take x = 13.

The first integer is 13.

The second integer is x + 1 = 13 + 1 = 14.

Result:

The two consecutive positive integers are 13 and 14.
© Kaliyuga Ekalavya. All rights reserved.

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