Question

The sum of the real solutions of the equation $5^{2x^2}-{2.5}^{x^2+x+1}-{3.5}^{2x+3}=0$, is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $1$

$5^{2x^2}-{2.5}^{x^2+x+1}-{3.5}^{2x+3}=0$
Dividing the expression by $5^{2x}$,
$5^{2x^2-2x}-{10.5}^{x^2-x}-{3.5}^3=0$
Let, $5^{x^2-x}=t$
On substitution,
$t^2-10t-375=0$
$(t-25)(t+15)=0\Rightarrow t=25,-15$
But, $t>0$, hence we reject $t=-15$
$\therefore 5^{(x^2-x)}=5^2\Rightarrow x^2-x=2$
$x^2-x-2=0\Rightarrow (x-2)(x+1)=0$
$x=2,-1$
$\therefore$ Sum of the real roots is $=1$.