6z+5(-2z-7)z=1

Solution for 6z+5(-2z-7)z=1 equation:

6z+5(-2z-7)z=1

We move all terms to the left:

6z+5(-2z-7)z-(1)=0

We multiply parentheses

-10z^2+6z-35z-1=0

We add all the numbers together, and all the variables

-10z^2-29z-1=0

a = -10; b = -29; c = -1;
Δ = b2-4ac
Δ = -292-4·(-10)·(-1)
Δ = 801
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{801}=\sqrt{9*89}=\sqrt{9}*\sqrt{89}=3\sqrt{89}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-29)-3\sqrt{89}}{2*-10}=\frac{29-3\sqrt{89}}{-20}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-29)+3\sqrt{89}}{2*-10}=\frac{29+3\sqrt{89}}{-20}
$