Question

six years before the age of mother was equal to the square of her son's Age 3 years h e n c e age will be thrice the age of her son then find the present ages of mother and son

Answer 1

Answer:

Age of mother is 54 years and age of son is 16 years!

Step-by-step explanation:

Let the present age of mother be "x"

And the present age of son be "y"

Then according to question,

Six years ago!

$\bold{\Longrightarrow\:x-6=(y-6)^2}$

$\bold{\Longrightarrow\:x-6=(y^{2}+6^{2}-2\times\:y\times6)}$

$\bold{{(a-b)^2\:=a^{2}+b^{2}-2ab}}$

$\bold{\Longrightarrow\:x-6=(y^{2}+36-12\times\:y)}$

$\bold{\Longrightarrow\:x=(y^{2}+36+6-12\times\:y)}$

$\bold{\Longrightarrow\:x=(y^{2}+42-12\times\:y)}$ ....(i)

Three years hence!

$\bold{\Longrightarrow\:x+3=(y+3)\times3}$

$\bold{\Longrightarrow\:x+3=3y+9}$

$\bold{\Longrightarrow\:x=3y+6}$ ....(ii)

∴$\bold{\Longrightarrow\:\underbrace{(y^{2}+42-12\timesy)}_{From\:eq^{n}\:(i)}=\underbrace{3y+6}_{From\:eq^{n}\:(ii)}}$

∴$\bold{\Longrightarrow\:y^{2}+36-15y=0}$

$\bold{\Longrightarrow\:y^{2}-15y+36=0}$ {This is a Quadratic Equation!}

By Quadratic Formula,

$\bold{y\:=\:\frac{-b\pm\sqrt{b^{2}-4\timesa\timesc}}{2\timesa}}$

$\bold{y\:=\:\frac{-(-15)\pm\sqrt{15^{2}-4\times1\times36}}{2\times1}}$

$\bold{y\:=\:\frac{+15\pm\sqrt{225-144}}{2}}$

$\bold{y\:=\:\frac{+15\pm\sqrt{81}}{2}}$

$\bold{y\:=\:\frac{+15\pm9}{2}}$

Taking value "+"

$\bold{y\:=\:\frac{+15+9}{2}}$

$\bold{\impliesy\:=\:\frac{+24}{2}}$

$\bold{\impliesy\:=\:12}$

Taking value "-"

But age cannot be in "-", So Value of "-" is neglected.

So, y = 12

And, We know that, $\bold{\Longrightarrow\:x=3y+6}$

∴x = 3(12)+6

    = 36+6

    = 42.

∴ Age of mother is 42 years and age of son is 12 years!

Thanks!

Answer 2

Answer:

Son = 12 years & Mother = 42 years

Step By Step Explanation:

6 years ago, Let age of son be x years.

Then, age of mother = x² years.

Now,

Present ages :

Age of Son = x + 6

Age of Mother = x² + 6

Again,

3 years hence :

Age of Son = x + 6 + 3 = x + 9

Age of Mother =  x² + 6 + 3 =  x² + 9

According To The Condition :

x²  + 9 = 3 (x + 9)

⇒ x²  + 9 = 3x + 27

⇒ x²  - 3x = 27 - 9

⇒ x² - 3x - 18 = 0

⇒ (x - 6) (x + 3) = 0

⇒ x = 6 or x = - 3

⇒ x = 6 ( Always take positive value )

Finally,

Finding their present ages :

Age of Son = x + 6 = 6 + 6 = 12 years.

Age of Mother =  x²  + 6 = (6)² + 6 = 42 years.

Hence, The present ages of mother and son is 12 years and 42 years respectively.