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-
$\dfrac1{\mathrm{a\alpha}+\mathrm b}+\dfrac1{\mathrm{\mathrm\alpha\mathrm\beta}+\mathrm b}$

Answer 1

This is a question of quadratic equations. First we will take the LCM of the denominators, take the terms of a and b common so that we can write the roots of the equation in terms of their product and the sum. Then we have to use the following formulas to simplify those expressions-
$\mathrm\alpha+\mathrm\beta=-\dfrac{\mathrm b}{\mathrm a}\\\\\mathrm{\alpha\beta}=\dfrac{\mathrm c}{\mathrm a}$

Complete step-by-step answer :
We have to convert the given expression such that it can be expressed in the form of $\mathrm\alpha+\mathrm\beta\;\mathrm{and}\;\mathrm{\alpha\beta}$ only, so that we can apply the given formulas, substitute the values and find the result.
So we will convert the equation as follows-
$\dfrac1{\mathrm{a\alpha}+\mathrm b}+\dfrac1{\mathrm{\mathrm\alpha\mathrm\beta}+\mathrm b}\\\=\dfrac{\left(\mathrm{\mathrm\alpha\mathrm\beta}+\mathrm b\right)+\left(\mathrm{a\alpha}+\mathrm b\right)}{\left(\mathrm{a\alpha}+\mathrm b\right)\left(\mathrm{\mathrm\alpha\mathrm\beta}+\mathrm b\right)}\\\=\dfrac{\mathrm a\left(\mathrm\beta+\mathrm\alpha\right)+2\mathrm b}{\mathrm a^2\mathrm{\alpha\beta}+\mathrm{ab}\left(\mathrm\alpha+\mathrm\beta\right)+\mathrm b^2}$
Now we will substitute the given formula in this expression to find its value using-
$\mathrm\alpha+\mathrm\beta=-\dfrac{\mathrm b}{\mathrm a}\\\\\mathrm{\alpha\beta}=\dfrac{\mathrm c}{\mathrm a}$
$=\dfrac{\mathrm a\left({\displaystyle\dfrac{-\mathrm b}{\mathrm a}}\right)+2\mathrm b}{\mathrm a^2\left({\displaystyle\dfrac{\mathrm c}{\mathrm a}}\right)+\mathrm{ab}\left({\displaystyle\dfrac{-\mathrm b}{\mathrm a}}\right)+\mathrm b^2}\\\=\dfrac{-\mathrm b+2\mathrm b}{\mathrm{ac}-\mathrm b^2+\mathrm b^2}\\\=\dfrac{\mathrm b}{\mathrm{ac}}$
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 $x^2$. A common mistake by students is that the students might get confused between the terms a, b and $\alpha, \beta$