10x(9-x)=x+10(9-x)+27

Solution for 10x(9-x)=x+10(9-x)+27 equation:

10x(9-x)=x+10(9-x)+27

We move all terms to the left:

10x(9-x)-(x+10(9-x)+27)=0

We add all the numbers together, and all the variables

10x(-1x+9)-(x+10(-1x+9)+27)=0

We multiply parentheses

-10x^2+90x-(x+10(-1x+9)+27)=0


We calculate terms in parentheses: -(x+10(-1x+9)+27), so:

x+10(-1x+9)+27

We multiply parentheses

x-10x+90+27

We add all the numbers together, and all the variables

-9x+117

Back to the equation:

-(-9x+117)


We get rid of parentheses

-10x^2+90x+9x-117=0

We add all the numbers together, and all the variables

-10x^2+99x-117=0

a = -10; b = 99; c = -117;
Δ = b2-4ac
Δ = 992-4·(-10)·(-117)
Δ = 5121
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:

$\sqrt{\Delta}=\sqrt{5121}=\sqrt{9*569}=\sqrt{9}*\sqrt{569}=3\sqrt{569}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(99)-3\sqrt{569}}{2*-10}=\frac{-99-3\sqrt{569}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(99)+3\sqrt{569}}{2*-10}=\frac{-99+3\sqrt{569}}{-20}
$