Question

The sum of a number and its reciprocal is ${\displaystyle \frac{17}{4}}$. The number is ……………………….
A. $5$
B. $4$
C. $6$
D. $3$

Answer 1

Hint: let the number be so its reciprocal will become and by putting these values in the original equation. We will get a quadratic equation and then by solving the equation we will get our required answer.]

Step by step solution:
Let the number be $$x$$.
So its reciprocal will be become $${\displaystyle \frac{1}{x}}$$
And as per the question $$x+{\displaystyle \frac{1}{x}}={\displaystyle \frac{17}{4}}$$
Now, we will try to form a quadratic equation
$$\begin{array}{l}\Rightarrow {\displaystyle \frac{x^2+1}{x}}={\displaystyle \frac{17}{4}}\\ \Rightarrow 4x^2-17x+4=0\end{array}$$
So, we get our quadratic equation. Now, we will try to find the value of x by factoring it
$$4x^2-16x-x+4=0$$
Taking $$4x$$ common from the first part and $$-1$$ common from the second part.
$$\begin{array}{l}\Rightarrow 4x\left(x-4\right)-1\left(x-4\right)=0\\ \Rightarrow \left(4x-1\right)\left(x-4\right)=0\end{array}$$
Now, by comparing the value we will get the values of x
$$x={\displaystyle \frac{1}{4}},x=4$$

So, our required answer is $$x={\displaystyle \frac{1}{4}},x=4$$

Note:
While letting the number in this type of question we need to take care that the value of x should not be zero otherwise we will get the value of $${\displaystyle \frac{1}{x}}as{\displaystyle \frac{1}{0}}$$ or $$\mathrm{\infty }$$ and also while finding the roots we must consider both the values that is $$+$$as well as $$-$$. For finding the roots we can also find one formula which is known as the sridharacharya formula in which we get both the roots of the equation. We can also find the quadratic equations if the roots are given . In that case we only need a sum of roots as well as a product of roots.