Question

If the roots of the equation $12{{x}^{2}}+mx+5=0$ are in the ratio 3: 2 the value of m is?
( a ) $5\sqrt{10}$
( b ) $3\sqrt{10}$
( c ) $2\sqrt{10}$
( d ) none of these

Answer 1

Hint: In this question, a quadratic equation is given with their roots and we have to find the value of m. as the roots are in ratio so we suppose the variable a with the ratios, and then we apply the conditions for the sum and the product of roots and after solving it, we get the value of m.

Formula used:
Sum of roots = $\dfrac{-b}{a}$
Product of roots = $\dfrac{c}{a}$

Complete step by step Solution:
Given equation is $12{{x}^{2}}+mx+5=0$
Compare the above equation with the standard form of quadratic equation $a{{x}^{2}}+bx+c=0$, we get
a = 12 , b = m and c = 5
And their roots are in the ratio 3 : 2
Let the roots of the equation are 3a and 2a
Now we apply the condition for the sum and the product of roots, and we get
Sum of roots ( 3a + 2a) = $-\dfrac{b}{a}$= $-\dfrac{m}{12}$
That is 5a = $-\dfrac{m}{12}$
     And m = - 60a
And the product of roots ( 3a $\times $ 2a) = $\dfrac{c}{a}$= $\dfrac{5}{12}$
That is 6${{a}^{2}}$= $\dfrac{5}{12}$
                ${{a}^{2}}$= $\dfrac{5}{72}$
That is a = $\pm \dfrac{\sqrt{5}}{6\sqrt{2}}$
Now put the value of a in m = -60 a, and we get
m = -60 $\times $$\pm \dfrac{\sqrt{5}}{6\sqrt{2}}$
By solving the above equation, we get
m = $\pm 5\sqrt{10}$

Therefore, the correct option is (a).

Note:We also find out the sum and the product of roots by simply putting the formula if x and y are the roots of any quadratic equation the value of xy will be equal to $\dfrac{ constant\, term}{coefficient\, of\, x^2}$ and sum of the roots that is x + y is equal to $\dfrac{ -coefficient\, of \,x}{coefficient\, of\, x^2}$ and by solving it we get the desired answer.