(3x+7)(5x-2)=61

Solution for (3x+7)(5x-2)=61 equation:

(3x+7)(5x-2)=61

We move all terms to the left:

(3x+7)(5x-2)-(61)=0

We multiply parentheses ..

(+15x^2-6x+35x-14)-61=0

We get rid of parentheses

15x^2-6x+35x-14-61=0

We add all the numbers together, and all the variables

15x^2+29x-75=0

a = 15; b = 29; c = -75;
Δ = b2-4ac
Δ = 292-4·15·(-75)
Δ = 5341
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{5341}=\sqrt{49*109}=\sqrt{49}*\sqrt{109}=7\sqrt{109}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(29)-7\sqrt{109}}{2*15}=\frac{-29-7\sqrt{109}}{30}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(29)+7\sqrt{109}}{2*15}=\frac{-29+7\sqrt{109}}{30}
$