15(15+2z+1)=(z+3)(6+z+3)

Solution for 15(15+2z+1)=(z+3)(6+z+3) equation:

15(15+2z+1)=(z+3)(6+z+3)

We move all terms to the left:

15(15+2z+1)-((z+3)(6+z+3))=0

We add all the numbers together, and all the variables

15(2z+16)-((z+3)(z+9))=0

We multiply parentheses

30z-((z+3)(z+9))+240=0

We multiply parentheses ..

-((+z^2+9z+3z+27))+30z+240=0


We calculate terms in parentheses: -((+z^2+9z+3z+27)), so:

(+z^2+9z+3z+27)

We get rid of parentheses

z^2+9z+3z+27

We add all the numbers together, and all the variables

z^2+12z+27

Back to the equation:

-(z^2+12z+27)


We add all the numbers together, and all the variables

30z-(z^2+12z+27)+240=0

We get rid of parentheses

-z^2+30z-12z-27+240=0

We add all the numbers together, and all the variables

-1z^2+18z+213=0

a = -1; b = 18; c = +213;
Δ = b2-4ac
Δ = 182-4·(-1)·213
Δ = 1176
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:

$\sqrt{\Delta}=\sqrt{1176}=\sqrt{196*6}=\sqrt{196}*\sqrt{6}=14\sqrt{6}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-14\sqrt{6}}{2*-1}=\frac{-18-14\sqrt{6}}{-2}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+14\sqrt{6}}{2*-1}=\frac{-18+14\sqrt{6}}{-2}
$