Question

If $m,n$ are the roots of the equation $x^2+7x-8=0,$ then the equation equation whose roots are $5m-11,5n-11$ is

• A. $x^2-57x+306=0$
• B. $x^2+306x-57=0$
• C. $x^2+57x+306=0$
• D. $x^2+57x-306=0$

Answer 1

The correct option is C $x^2+57x+306=0$

Given: $x^2+7x-8=0;$ roots are $m,n$

$\text{Sum of roots}=m+n=-7\cdots \left(i\right)$

$\text{Product of roots}=m.n=-8\cdots \left(ii\right)$

Now we have to find the equation whose roots are $5m-11,5n-11$

$\therefore \text{Sum of roots}=(5m-11)+(5n-11)=\frac{-b}{a}$

$\Rightarrow \text{Sum of roots}=5(m+n)-22=\frac{-b}{a}$

$\Rightarrow \text{Sum of roots}=\frac{-b}{a}=5(-7)-22\left[\text{From}\right(i),(ii\left)\right]$

$\Rightarrow \text{Sum of roots}=\frac{-b}{a}=-57\cdots \left(iii\right)$

Now, $\text{Product of roots}=(5m-11).(5n-11)=\frac{c}{a}$

$\Rightarrow \text{Productof roots}=\frac{c}{a}=25m.n-55(m+n)+121$

$\Rightarrow \text{Productof roots}=\frac{c}{a}=25(-8)-55(-7)+121\left[\text{From}\right(i),(ii\left)\right]$

$\Rightarrow \text{Productof roots}=\frac{c}{a}=306\cdots \left(iv\right)$

Now we have the sum , products of roots for the required equation.And we can write the equation as: $x^2-\left(\text{sum of roots}\right)x+\left(\text{product of roots}\right)=0$

$\therefore x^2-(-57)x+\left(306\right)=0$

$\Rightarrow x^2+57x+306=0$