Ashu is x years old while his mother Mrs Veena is ${{x}^{2}}$ years old. Five years hence Mrs Veena will be three times old as Ashu. The product of their present ages is:

Question

Ashu is x years old while his mother Mrs Veena is ${{x}^{2}}$  years old. Five years hence Mrs Veena will be three times old as Ashu. The product of their present ages is:

Vedantu Content Team · Accepted Answer

Hint: Given age in terms of $x$ may be in any degree of polynomial, but thereafter the age will always increase linearly i.e. as years pass by the age increase by 1 every year irrespective of it being $x,{{x}^{2}},{{x}^{3}}$ now at present. Here Ashu’s and Mrs. Veena’s age also will be just 5 more after 5 years irrespective of $x,{{x}^{2}}$ both of them will increase by 5.

Complete step-by-step answer:

Given in the question the present ages of both is

Ashu age is $x$ years

Her mother's age is ${{x}^{2}}$ years

After 5 years from now, the ages of both of them will be:

Ashu after 5 years = $x+5$ years

Her mother after 5 years = ${{x}^{2}}+5$ years

Given in the question that five years hence the age of her mother will be 3 times her age after 5 years.

By using above condition, we get a polynomial as follows:

${{x}^{2}}+5=3\left( x+5 \right)$

By simplifying above equation, we convert it into the form:

${{x}^{2}}+5=3x+15$

By subtracting with \[\left( 3x+15 \right)\] on both sides of equation, we get:

${{x}^{2}}+5-3x-15=0$

By simplifying above equation, we convert it into form:

${{x}^{2}}-3x-10=0$

By writing $-3x=-5x+2x$ , we convert the equation into form:

${{x}^{2}}-5x+2x-10=0$

By taking $x$ common in first two terms and 2 common in next two terms, we convert the equation into:

$x\left( x-5 \right)+2\left( x-5 \right)=10$

By taking $\left( x-5 \right)$ common in the equation, we convert it into:

$\left( x-2 \right)\left( x-5 \right)=0$

By equating each term to zero, we get its roots as:

$\begin{align}

& x+2=0;\text{ }x-5=0 \\

& x=-2;\text{ }x=5 \\

\end{align}$

As age cannot be negative, we get: $x=5$

Present age of Ashu = $x$ = 5 years

Present age of Mother = ${{x}^{2}}$ =25 years

Product of their ages is 5 x 25 = 125

Therefore 125 is the product of their both ages at present.

Note: Be careful while solving quadratic equations, as the $x$ here in this case is age, we need to remember $x > 0$ always, as age is always positive.