X^2-6x=1/2+42

Solution for X^2-6x=1/2+42 equation:

X^2-6X=1/2+42

We move all terms to the left:

X^2-6X-(1/2+42)=0

We get rid of parentheses

X^2-6X-42-1/2=0

We multiply all the terms by the denominator


X^2*2-6X*2-1-42*2=0

We add all the numbers together, and all the variables

X^2*2-6X*2-85=0

Wy multiply elements

2X^2-12X-85=0

a = 2; b = -12; c = -85;
Δ = b2-4ac
Δ = -122-4·2·(-85)
Δ = 824
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:

$\sqrt{\Delta}=\sqrt{824}=\sqrt{4*206}=\sqrt{4}*\sqrt{206}=2\sqrt{206}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{206}}{2*2}=\frac{12-2\sqrt{206}}{4}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{206}}{2*2}=\frac{12+2\sqrt{206}}{4}
$