1/2+x(x-15)+1/2x+(x-25)+100=540

Solution for 1/2+x(x-15)+1/2x+(x-25)+100=540 equation:

1/2+x(x-15)+1/2x+(x-25)+100=540

We move all terms to the left:

1/2+x(x-15)+1/2x+(x-25)+100-(540)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


determiningTheFunctionDomain
x(x-15)+1/2x+(x-25)+100-540+1/2=0

We add all the numbers together, and all the variables

x(x-15)+1/2x+(x-25)-440+1/2=0

We multiply parentheses

x^2-15x+1/2x+(x-25)-440+1/2=0

We get rid of parentheses

x^2-15x+1/2x+x-25-440+1/2=0

We calculate fractions


x^2-15x+x-25-440=0

We add all the numbers together, and all the variables

x^2-14x-465=0

a = 1; b = -14; c = -465;
Δ = b2-4ac
Δ = -142-4·1·(-465)
Δ = 2056
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:

$\sqrt{\Delta}=\sqrt{2056}=\sqrt{4*514}=\sqrt{4}*\sqrt{514}=2\sqrt{514}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{514}}{2*1}=\frac{14-2\sqrt{514}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{514}}{2*1}=\frac{14+2\sqrt{514}}{2}
$