Question

If $\mathrm\alpha\;\mathrm{and}\;\mathrm\beta$ are the zeroes of the quadratic polynomial $f(x) = ax^2+ bx + c$, then evaluate the following-
$\dfrac1{\mathrm\alpha}+\dfrac1{\mathrm\beta}-2\mathrm{\mathrm\alpha\mathrm\beta}$

Answer 1

This is a question of quadratic equations. We will first simplify the term $\dfrac{1}{\alpha } + \dfrac{1}{\beta }$ in the form of the sum ($\alpha + \beta$) and the product of the roots $\alpha \beta$. Once we do that, we can substitute the values of the sum and the product of the roots of the quadratic equation $f(x) = ax^2+ bx + c$ respectively using the formulas-
$\mathrm\alpha+\mathrm\beta=-\dfrac{\mathrm b}{\mathrm a}\\\\\mathrm{\mathrm\alpha\mathrm\beta}=\dfrac{\mathrm c}{\mathrm a}$

Complete step-by-step answer :
We have to convert the given expression such that it can be expressed in the form of $\mathrm\alpha+\mathrm\beta\;\mathrm{and}\;\mathrm{\mathrm\alpha\mathrm\beta}$ only, so that we can apply the given formulas, substitute the values and find the result.
So we will convert the equation by taking the LCM as follows-
$\dfrac1{\mathrm\alpha}+\dfrac1{\mathrm\beta}-2\mathrm{\mathrm\alpha\mathrm\beta}\\\=\dfrac{\mathrm\beta+\mathrm\alpha}{\mathrm{\mathrm\alpha\mathrm\beta}}-2\mathrm{\mathrm\alpha\mathrm\beta}$
We have now converted the expression in terms of the sum and the product of the roots of the quadratic equation. Now, we will simplify this by applying the formula for the relationship between the roots and the coefficients of the equation. So, we will substitute the given formula in this expression to find its value using-
$\mathrm\alpha+\mathrm\beta=-\dfrac{\mathrm b}{\mathrm a}\\\\\mathrm{\mathrm\alpha\mathrm\beta}=\dfrac{\mathrm c}{\mathrm a}$
$=\dfrac{-\left({\displaystyle\dfrac{\mathrm b}{\mathrm a}}\right)}{\displaystyle\dfrac{\mathrm c}{\mathrm a}}-2\dfrac{\mathrm c}{\mathrm a}\\\=-\dfrac{\mathrm b}{\mathrm c}-\dfrac{2\mathrm c}{\mathrm a}\\\=\dfrac{-\left(\mathrm{ab}+2\mathrm c^2\right)}{\mathrm{ac}}$
This is the required answer.

Note :The above given formula signifies that the sum of roots of a quadratic equation is the negative of the ratio of coefficient of $x$ and $x^2$. Also, the product of roots is the ratio of constant term and coefficient of $x^2$. It is also recommended to simplify the final answer.