Question

3. a) The sum of two numbers is 18 and their product is 45, find the numbers.​

Answer 1

• The numbers are 3 and 15 OR 15 and 3.

Given :

• The sum of two numbers = 18.
• The product of both numbers = 45.

Go Find :

• The numbers.

Solution :

Let,

The first number be x.

The second number be y.

According to the question,

• x + y = 18 ––––––(1)

• xy = 45 ––––––(2)

From equation (1),

$:\implies \rm x + y = 18 \\ \\ :\implies \rm x= 18 - y \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: (3)$

Substitue the. equation (3) in equation (2),

$:\implies \rm xy = 45 \\ \\ :\implies \rm (18 - y)y = 45 \\ \\ :\implies \rm 18y - {y}^{2} = 45 \\ \\ :\implies \rm {y}^{2} - 18y + 45 = 0 \\ \\ \sf{By \: Splitting \: The \: Middle \: Term} \\ \\ :\implies \rm {y}^{2} - 15y - 3y + 45 = 0 \\ \\ :\implies \rm y(y - 15) - 3(y - 15) = 0 \\ \\ :\implies \rm (y - 3) \: (y - 15) = 0 \\ \\ :\implies \rm (y - 3) = 0 \: ; \: (y - 15) = 0 \\ \\ :\implies \rm y - 3 = 0 \: ; \: y - 15 = 0 \\ \\ :\implies \rm y = 3 \: ; \: y = 15$

Hence,

The value of y is 3 or 15.

So, the second number is 3 or 15.

Now, substitute both the values of y in equation (1),

$1 ) \: \: \: y = 3 \\ \\ :\implies \rm x + y = 18 \\ \\ :\implies \rm x + 3 = 18 \\ \\ :\implies \rm x = 18 - 3 \\ \\ :\implies \rm x = 15 \\ \\ \\ \\ 2) \: \: \: y = 15 \\ \\ :\implies \rm x + y = 18 \\ \\ :\implies \rm x + 15 = 18 \\ \\ :\implies \rm x = 18 - 15 \\ \\ :\implies \rm x = 3$

Hence,

The value of x is 15 or 3.

So, the first number is 15 or 3.

Hence,

The numbers are 3 & 15 or 15 & 3.