Question

Find the present age of a man if his age is 40 years hence will become equal to the square of what his age was 32 years ago.

Answer 1

To obtain the present age of the man we will let it be $x$. Then by using the information given we will form a relation between his age 40 years hence and 32 years ago. Finally solve the obtained equation by using quadratic formula i.e. $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.

Complete step-by-step solution:
Let us take the present age of the man as:
Present age $=x$
So his age 40 years hence will be:
After 40 years $=\left( x+40 \right)$……..$\left( 1 \right)$
Next, his age 32 years ago will be:
Before 32 years $=\left( x-32 \right)$…….$\left( 2 \right)$
Now, from the question we get that after 40 years age is equal to square of age 32 years ago so,
$\Rightarrow \left( x+40 \right)={{\left( x-32 \right)}^{2}}$
We will simplify the above equation and get,
$\begin{aligned} & \Rightarrow x+40={{\left( x-32 \right)}^{2}} \\\ & \Rightarrow x+40={{x}^{2}}-2\times x\times 32+{{32}^{2}} \\\ & \Rightarrow 0={{x}^{2}}-64x+1024-x-40 \\\ \end{aligned}$
$\therefore {{x}^{2}}-65x+984=0$…….$\left( 2 \right)$
Next, we will use the quadratic formula for the quadratic equation obtained above.
For any quadratic equation of form$f\left( x \right)=a{{x}^{2}}+bx+c$
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$……..$\left( 3 \right)$
On comparing from equation (2) we get,
$\begin{aligned} & a=1 \\\ & b=-65 \\\ & c=984 \\\ \end{aligned}$
Substituting the above value in equation 93) we get,
$\begin{aligned} & \Rightarrow x=\dfrac{-\left( -65 \right)\pm \sqrt{{{\left( -65 \right)}^{2}}-4\times 1\times 984}}{2\times 1} \\\ & \Rightarrow x=\dfrac{65\pm \sqrt{4225-3936}}{2} \\\ & \Rightarrow x=\dfrac{65\pm \sqrt{289}}{2} \\\ & \Rightarrow x=\dfrac{65\pm 17}{2} \\\ \end{aligned}$
So we get two values as,
$\begin{aligned} & \Rightarrow x=\dfrac{65+17}{2} \\\ & \Rightarrow x=\dfrac{82}{2} \\\ & \therefore x=41 \\\ \end{aligned}$
And
$\begin{aligned} & \Rightarrow x=\dfrac{65-17}{2} \\\ & \Rightarrow x=\dfrac{48}{2} \\\ & \therefore x=24 \\\ \end{aligned}$
The present age can’t be less than 40 years.
Hence, the present age of the man is 41 years.

Note: The quadratic formula for quadratic function is used widely as it is easy from using hit and trial or using factoring methods when the numbers involved are big. An equation can have more than one solution but depending on the question we have to filter out the one which is not possible as in this case the age of the man can’t be less than 40 years.