Question

The product of ages of Sachin $8$ years ago and $2$ years later is $1200$. Find the present age of Sachin.

Answer 1

Hint: Assume the age of Sachin to be x years. Now, to find the age of Sachin $8$ years ago, subtract it from the present age of Sachin and to find the age of Sachin $2$ years later, add it to the present age of Sachin. Then form a quadratic equation from the first statement and solve it to find the value of x.

Complete step-by-step answer:
Let the present age of Sachin be x years. Then to find the age of Sachin $8$ years ago, subtract it from the present age of Sachin and to find the age of Sachin $2$ years later, add it to the present age of Sachin.
So the age of Sachin $8$ years ago will be $x - 8$ yrs and the age of Sachin $2$ years later will be $x + 2$ yrs.
Now, according to the question, it is given that the product of ages of Sachin $8$ years ago and $2$ years later is $1200$ so we can write-
$ \Rightarrow \left( {x - 8} \right)\left( {x + 2} \right) = 1200$
On solving, we get-
$ \Rightarrow x\left( {x + 2} \right) - 8\left( {x + 2} \right) = 1200$
On multiplication of the above terms, we get-
$ \Rightarrow {x^2} + 2x - 8x - 16 = 1200$
On subtracting the second from the third term, we get-
$ \Rightarrow {x^2} - 6x - 16 = 1200$
On transferring the constant term on the left side, we get-
$ \Rightarrow {x^2} - 6x - 16 - 1200 = 0$
On solving, we have-
$ \Rightarrow {x^2} - 6x - 1216 = 0$
This is a quadratic equation. We can solve it by factoring the equation to find the value of x.
On factorization, we get-
$ \Rightarrow {x^2} - 38x + 32x - 1216 = 0$
On taking common from first and second term and third and fourth term, we get-
$ \Rightarrow x\left( {x - 38} \right) + 32\left( {x - 38} \right) = 0$
On taking the term $\left( {x - 38} \right)$common, we get-
$ \Rightarrow \left( {x - 38} \right)\left( {x + 32} \right) = 0$
Now, on equating either of the multiplication terms to zero we get-
$ \Rightarrow \left( {x - 38} \right) = 0$or$\left( {x + 32} \right) = 0$
On solving we get-
$ \Rightarrow x = 38$ or $x = - 32$
Since we know that age of a person cannot be negative hence the present age of Sachin is $38$ yrs.
The present age of Sachin is $38$ yrs.
Note: Here we can also solve the quadratic equation using the discriminant method. If the equation in the standard form $a{x^2} + bx + c = 0$ then we find the value of x using formula-
$ \Rightarrow $ $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
So, on comparing the standard equation with the eq. given below-
$ \Rightarrow {x^2} - 6x - 1216 = 0$
We get-
$ \Rightarrow $ a=$1$ , b=$ - 6$ and c=$ - 1216$
Then putting the values in the formula, we get-
$ \Rightarrow $ $x = \dfrac{{ - \left( { - 6} \right) \pm \sqrt {{{\left( { - 6} \right)}^2} - 4 \times 1 \times \left( { - 1216} \right)} }}{2}$
On simplifying, we get-
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {36 + 4864} }}{2}$
On simplifying further,
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {4900} }}{2} = \dfrac{{6 \pm 70}}{2}$
On division, we get-
$ \Rightarrow x = 3 \pm 35$
On solving, we get-
$ \Rightarrow x = 38$ or $x = - 32$