Question

If the roots of the quadratic equations \[h{x^2} + 21x + 10 = 0\left( {b \ne 0} \right)\] are in the ratio $2:5$. Find the possible values of $h$.

Answer 1

Hint: Most importantly we will discover the number of roots and the results of roots. And by using the equation we will find out the sum of zeroes which will be $\dfrac{{ - b}}{a}$ and the product of zeroes $\dfrac{c}{a}$. And on comparing the equation we can find out the value for the$h$.

Formula used:
As we know, when we have an equation $a{x^2} + bx + c = 0$
At that point, the sum of zeroes $ = \dfrac{{ - b}}{a}$
Also the product of zeroes $ = \dfrac{c}{a}$
Here,
$a,b,c$, are the coefficients.

Complete step-by-step answer:
We have the ratios given as $2:5$
So let us assume that the roots of the ratios are $2x$ and $5x$
Then, the sum of roots will be equals to $2x + 5x = 7x$
And the product of the roots will be $2x \times 5x = 10{x^2}$
As we've got found within the formula,
For the condition \[h{x^2} + 21x + 10 = 0\]
At that point, the sum of zeroes $ = \dfrac{{ - 21}}{h}$
And also the product of zeroes $ = \dfrac{{10}}{h}$
As we already know the sum of roots will be equals to $7x$
And the product of the roots will be $10{x^2}$
Therefore,
$ \Rightarrow 7x = \dfrac{{ - 21}}{h}$
What's more, from here, we will discover the estimation of $x$
$ \Rightarrow x = \dfrac{{ - 3}}{h}$
Now on comparing the product of zeroes, we get
$ \Rightarrow 10{x^2} = \dfrac{{10}}{h}$
And now on substituting the values of$x$, we get
$ \Rightarrow 10{\left( {\dfrac{{ - 3}}{h}} \right)^2} = \dfrac{{10}}{h}$
Now on solving for the values of$h$, we get
$ \Rightarrow \dfrac{9}{{{h^2}}} = \dfrac{1}{h}$
And from here, we get $h = 9$
Therefore, $9$ will be the possible values of $h$.


Note: So for solving this type of question, the only thing we should know is how we can calculate the product and sum of zeroes by using the formula and with a few calculations we can find the required values. In this way, we will solve it comfortably.