Question

Find the values of $m$ for which the quadrilateral equation $x^2-m(2x-8)-15=0$ has
(i) equal roots
(ii) both roots positive.

Answer 1

Given eq. is;

$f\left(x\right)=x^2-2mx+8m-15=0$

1). For equal roots, $D=0$

$\Rightarrow$$D=b^2-4ac=0$

$\Rightarrow$ $4m^2-32m+60=0$

$\Rightarrow$ $m^2-8m+15=0$

$\Rightarrow$ $m=3,5$

2). For both roots positive, $D>=0$ & $f\left(0\right)>0$

$f\left(0\right)=8m-15>0$

$\Rightarrow$ $m>\frac{15}{8}$ ........(1).

$D>=0$

$\Rightarrow$ $m^2-8m+15>=0$

$\Rightarrow$ $(m-3)(m-5)>=0$ ..........(2).

From (1) & (2),

$\Rightarrow$ $m>=5$

$\Rightarrow$ $mϵ[5,\infty )$