NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 Question 3

NCERT Class X Chapter 4:Quadratic Equation Exercise 4.2 Question 3

Question:

Find two numbers whose sum is 27 and product is 182.

Given:

Sum of two numbers = 27
Product of two numbers = 182

To Find:

The two numbers.

Formula:

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 one of the numbers be 'x' and the other number be 'y'

⇒x + y = 27

⇒y = 27 - x

According to the given condition, their product is 182:

⇒ x × y = 27

⇒ x × (27 - x) = 182

⇒ 27x - x2 = 182
Rearrange the equation into the standard quadratic form ax2 + bx + c = 0:

⇒ x2 - 27x + 182 = 0
Now, we solve this quadratic equation by factorization.

Here, a = 1, b = -27, c = 182.

Product (a × c) = 1 × 182 = 182.

Sum (b) = -27.
We need to find two numbers whose product is 182 and sum is -27.

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 and sum is -27; -27 < 0 both numbers are negative.

Therefore the two numbers are -13 and -14.
Rewrite the middle term (-27x) using these numbers:

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

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

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

Case 1:  x - 14 = 0

⇒ x = 14

If x = 14, the other number is 27 - 14 = 13.
Case 2: x - 13 = 0

⇒ x = 13

If x = 13, the other number is 27 - 13 = 14.

Result:

The two numbers are 13 and 14.
© Kaliyuga Ekalavya. All rights reserved.

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