5(7-z)=z(z-9)

Solution for 5(7-z)=z(z-9) equation:

5(7-z)=z(z-9)

We move all terms to the left:

5(7-z)-(z(z-9))=0

We add all the numbers together, and all the variables

5(-1z+7)-(z(z-9))=0

We multiply parentheses

-5z-(z(z-9))+35=0


We calculate terms in parentheses: -(z(z-9)), so:

z(z-9)

We multiply parentheses

z^2-9z

Back to the equation:

-(z^2-9z)


We get rid of parentheses

-z^2-5z+9z+35=0

We add all the numbers together, and all the variables

-1z^2+4z+35=0

a = -1; b = 4; c = +35;
Δ = b2-4ac
Δ = 42-4·(-1)·35
Δ = 156
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:

$\sqrt{\Delta}=\sqrt{156}=\sqrt{4*39}=\sqrt{4}*\sqrt{39}=2\sqrt{39}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{39}}{2*-1}=\frac{-4-2\sqrt{39}}{-2}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{39}}{2*-1}=\frac{-4+2\sqrt{39}}{-2}
$