Question

Rohan's mother is 26 years older than him. The product of their ages 3 years from now will be 360. Find Rohan's present age.

Answer 1

Hint: Now to solve the question we will consider the age of Rohan. Now we know the age difference between Rohan and his mother, so add that year in Rohan’s age to get his mother’s age. Now 3 years from now, the product of their age is given. It will give the quadratic equation. Now we will solve the equation to find the age of Rohan.

Complete step by step answer:
Given: - Rohan's mother is 26 years older than him. The product of their ages 3 years from now will be 360.
Let the present age of Rohan be $x$. So, Rohan’s mother’s age will be $\left( {x + 26} \right)$.
It is given that after 3 years from now, the product of Rohan's and his mother's ages will be 360 years.
$ \Rightarrow \left( {x + 3} \right)\left[ {\left( {x + 26} \right) + 3} \right] = 360$
Add the terms in the brackets,
$ \Rightarrow \left( {x + 3} \right)\left( {x + 29} \right) = 360$
Multiply the terms,
$ \Rightarrow {x^2} + 32x + 87 = 360$
Move 360 on the left side and subtract from 87,
$ \Rightarrow {x^2} + 32x - 273 = 0$
Factorization of the above quadratic equation using formula.
$x = $ $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Put a= 1, b= 32 and c= -273,
$ \Rightarrow x = \dfrac{{ - 32 \pm \sqrt {{{32}^2} - 4 \times 1 \times ( - 273)} }}{{2 \times 1}}$
Multiply and square the terms and add them in the square root,
$ \Rightarrow x = \dfrac{{ - 32 \pm \sqrt {2116} }}{2}$
Then,
$ \Rightarrow x = \dfrac{{ - 32 \pm 46}}{2}$
Then,
$ \Rightarrow x = - 39,7$
Since age cannot be negative.
$\therefore x = 7$

Hence, Rohan’s age is 7 years.

Note:
A quadratic equation is a polynomial equation of degree 2. A quadratic equation has two solutions. Either two distinct real solutions, one double real solution, or two imaginary solutions.
There are several methods you can use to solve a quadratic equation:
- Factoring
- Completing the Square
- Quadratic Formula
- Graphing
All methods start with setting the equation equal to zero.

Alternative method to solve the quadratic equation in the solution:
${x^2} + 32x - 273 = 0$
We can write 32 as (39-7)
$ \Rightarrow {x^2} + \left( {39 - 7} \right)x - 273 = 0$
Open the brackets and multiply the terms,
$ \Rightarrow {x^2} + 39x - 7x - 273 = 0$
Take common factors from the equation,
$ \Rightarrow x(x + 39) - 7(x + 39) = 0$
Take common factors from the equation,
$ \Rightarrow (x + 39)(x - 7) = 0$
Then,
$ \Rightarrow x = - 39,7$
Since age cannot be negative. So,
$\therefore x = 7$

Hence, Rohan’s age is 7 years.