Question

The sum of ages of two friends is 20 years. Four years ago the product of their ages was 48.show that these statements can not be true.

Answer 1

$\: \mathbf{Let \: \: present \: \: ages \: of \: friends \: are \: \: x \: years \: and \: \: ( 20 - x ) \: years, }$


$\bold{Four \: \: years \: \: ago, } \\ \bold{ Age \: \: of \: friends = ( x - 4 ) \: \: years \: and \: \: ( 20 - x - 4 ) = ( 16 - x ) \: years, }$



Product of their ages = ( x - 4 ) ( 16 - x )

=> 48 = ( x - 4 ) ( 16 - x )

=> 48 = 16x - x² - 64 + 4x

=> x² - 20x + 112 = 0


$\bold{ On \: comparing \: the \: given \: equation \: with ( ax ^{2} + bx + c = 0 ), \: we \: get}$


a = 1
b = - 20
c = 112



Hence, Discriminant = b² - 4ac

=> ( - 20 )² - 4( 112 × 1 )

=> 400 - 448

=> - 48








$\mathbf{- 48 \: has \: not \: any \: root, \: Hence \: this \: cant \: be \: used \: in \: quadratic \: equation, \: due \: to \: which \: the \: given \: statement \: is \: false} \:$

Answer 2

$\purple{\boxed{\huge Answer}}$

Given :-

Sum of age of two friends is 20 years.

Let ,

Age of first friend be x

Age of second friend =20-x

Four years ago:-

age of 1st friend = x - 4✔

Age of 2nd friend = (20 - x) - 4 =16 - x✔

Product of the age of two friends = 48 years :-

According to question

=> (x - 4) (16 - x) = 48

=> x (16 - x) - 4 (16 - x) = 48

=> 16x - x² - 64 + 4x = 48

=>- x² + 16x + 4x - 64 - 48= 0

=> - x² +20x - 112 = 0

=> x² - 20x + 112 = 0

Comparing the equation with ax² + bx + c = x² - 20x + 112

Here

a = 1

b = -20

c = 112

Now we will used discriminant formula :-

D = b² - 4ac

=> (-20)² - (4 × 1 × 112)

=> 400 - 448

=> - 48

(D < 0 )i.e D is less than 0 its mean it have no real roots

Hence,

It is not possible to find the present age