7^k*7^-2k-3=49

Solution for 7^k*7^-2k-3=49 equation:

7^k*7^-2k-3=49

We move all terms to the left:

7^k*7^-2k-3-(49)=0

We add all the numbers together, and all the variables

-2k+7^k*7^-52=0

Wy multiply elements

49k^2-2k-52=0

a = 49; b = -2; c = -52;
Δ = b2-4ac
Δ = -22-4·49·(-52)
Δ = 10196
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{10196}=\sqrt{4*2549}=\sqrt{4}*\sqrt{2549}=2\sqrt{2549}$
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{2549}}{2*49}=\frac{2-2\sqrt{2549}}{98}
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{2549}}{2*49}=\frac{2+2\sqrt{2549}}{98}
$