Question

expain the steps...why do we take 12/x....and what is the basic format of taking number in such kind of cases ..

Answer 1

My friend , It has been done to make the equation in quadratic . You can also use linear equation in two variables assuming variables as x and y . The answer will be same and also in any exam you are not bounded to use quadratic in these type of questions.

Answer 2

It has taken tens digit as x

let the unit digit be y.

given that their product is 12.

So x × y = 12

=> y = 12/x.

That's why in the book, it has directly taken unit digit as 12/x.

So the tens digit is x and unit digit is 12/x.

Then do you know what the number will be?
The number will be[ (10 × x) +(1 × 12/x) ] or (10x + 12/x).

Now, it is given that if you add 36 to it, the places of the digits change. That means after adding 36, tens digit will be 12/x and unit digit will x in the new number.
So new number will be [ (10 × 12/x) + (1×x) ] or (120/x + x).

Thus

$(10x + \frac{12}{x} ) + 36 = \frac{120}{x} + x \\ \\ 10x - x + \frac{12}{x} - \frac{120}{x} + 36 = 0$

$9x + \frac{12 - 120}{x} + 36 = 0 \\ \\ 9x - \frac{108}{x} + 36 = 0$

$\frac{9x \times x - 108 + 36 \times x}{x} = 0 \\ \\ \frac{9 {x}^{2} + 36x - 108 }{x} = 0$

$9 {x}^{2} + 36x - 108 = 0 \times x = 0$

$9 {x}^{2} + 36x - 108 = 0$

$9( {x}^{2} + 4x - 12) = 0 \\ \\ {x}^{2} + 4x - 12 = 0$

That's the required equation. If you solve it, you will get x and you can get the number once you know x.