x2-6x+1=3x+5/2

Solution for x2-6x+1=3x+5/2 equation:

x2-6x+1=3x+5/2

We move all terms to the left:

x2-6x+1-(3x+5/2)=0

We add all the numbers together, and all the variables

x2-6x-(+3x+5/2)+1=0

We add all the numbers together, and all the variables

x^2-6x-(+3x+5/2)+1=0

We get rid of parentheses

x^2-6x-3x+1-5/2=0

We multiply all the terms by the denominator


x^2*2-6x*2-3x*2-5+1*2=0

We add all the numbers together, and all the variables

x^2*2-6x*2-3x*2-3=0

Wy multiply elements

2x^2-12x-6x-3=0

We add all the numbers together, and all the variables

2x^2-18x-3=0

a = 2; b = -18; c = -3;
Δ = b2-4ac
Δ = -182-4·2·(-3)
Δ = 348
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:

$\sqrt{\Delta}=\sqrt{348}=\sqrt{4*87}=\sqrt{4}*\sqrt{87}=2\sqrt{87}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{87}}{2*2}=\frac{18-2\sqrt{87}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{87}}{2*2}=\frac{18+2\sqrt{87}}{4}
$