(x-4)(4+x)=54(x-5)(x-10)

Solution for (x-4)(4+x)=54(x-5)(x-10) equation:

(x-4)(4+x)=54(x-5)(x-10)

We move all terms to the left:

(x-4)(4+x)-(54(x-5)(x-10))=0

We add all the numbers together, and all the variables

(x-4)(x+4)-(54(x-5)(x-10))=0

We use the square of the difference formula

x^2-(54(x-5)(x-10))-16=0

We multiply parentheses ..

x^2-(54(+x^2-10x-5x+50))-16=0


We calculate terms in parentheses: -(54(+x^2-10x-5x+50)), so:

54(+x^2-10x-5x+50)

We multiply parentheses

54x^2-540x-270x+2700

We add all the numbers together, and all the variables

54x^2-810x+2700

Back to the equation:

-(54x^2-810x+2700)


We get rid of parentheses

x^2-54x^2+810x-2700-16=0

We add all the numbers together, and all the variables

-53x^2+810x-2716=0

a = -53; b = 810; c = -2716;
Δ = b2-4ac
Δ = 8102-4·(-53)·(-2716)
Δ = 80308
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:

$\sqrt{\Delta}=\sqrt{80308}=\sqrt{4*20077}=\sqrt{4}*\sqrt{20077}=2\sqrt{20077}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(810)-2\sqrt{20077}}{2*-53}=\frac{-810-2\sqrt{20077}}{-106}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(810)+2\sqrt{20077}}{2*-53}=\frac{-810+2\sqrt{20077}}{-106}
$