Question

sum of the present ages of two friends are 23 years, five years ago product of their ages was 42.find their ages 5 years hence.​

Answer 1

Let the present ages of two friends be : P and Q

★  Sum of present ages of two friends are 23

$\longrightarrow$  P + Q = 23

$\longrightarrow$  P = 23 - Q

Five years ago :

●  Age of first friend : P - 5

●  Age of second friend : Q - 5

★  Five years ago, Product of their ages was 42

$\longrightarrow$  (P - 5)(Q - 5) = 42

$\longrightarrow$  (23 - Q - 5)(Q - 5) = 42

$\longrightarrow$  (18 - Q)(Q - 5) = 42

$\longrightarrow$  18Q - 90 - Q² + 5Q = 42

$\longrightarrow$  Q² - 23Q + 132 = 0

$\longrightarrow$  Q² - 12Q - 11Q + 132 = 0

$\longrightarrow$  Q(Q - 12) - 11(Q - 12) = 0

$\longrightarrow$  (Q - 12)(Q - 11) = 0

$\longrightarrow$  Q = 12  or  Q = 11

$\longrightarrow$  If Q = 12 then P = 11  or  If Q = 11 then P = 12

5 years hence :

●  Age of P = 12 + 5 = 17  or  Age of P = 11 + 5 = 16

●  Age of Q = 11 + 5 = 16  or  Age of Q = 12 + 5 = 17

Answer 2

Sum Of Present Age=23

Let Present Age of one Friend=x years

Then Present Age ot other Friend=(23-x) years

5 years ago

One Friend Age was=(x-5) years

Other Friend Age was=(23-x-5)=(18-x) years

Product of Their age=42

(x-5)(18-x)=42

x(18-x)-5(18-x)=42

18x-x²-90+5x=42

23x-x²-90=42.

-x²+23x-90-42=0

By Middle Term Splitting

-x²+11x+12x-132=0

Moving Term to RHS

0=0-(-x²+11x+12x-132)

0=x²-11x-12x+132

0=X(x-11)-12(x-11)

0=(x-12)(x-11)

We got

x-12=0

x=0+12

x=12

And

x-11=0

x=0+11

x=11

Let Present Age of one Friend=12

Other friend Present Age=23-x

=23-12

=11

After 5 years

Age of one friend=12+5=17 years

Age of Other Friend=11+5=16 years

$\boxed{\mathbf{\huge{Ages=17,16\:years}}}$