x(x+1/2)=68

Solution for x(x+1/2)=68 equation:

x(x+1/2)=68

We move all terms to the left:

x(x+1/2)-(68)=0

We add all the numbers together, and all the variables

x(+x+1/2)-68=0

We multiply parentheses

x^2+x^2-68=0

We add all the numbers together, and all the variables

2x^2-68=0

a = 2; b = 0; c = -68;
Δ = b2-4ac
Δ = 02-4·2·(-68)
Δ = 544
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:

$\sqrt{\Delta}=\sqrt{544}=\sqrt{16*34}=\sqrt{16}*\sqrt{34}=4\sqrt{34}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-4\sqrt{34}}{2*2}=\frac{0-4\sqrt{34}}{4}
=-\frac{4\sqrt{34}}{4}
=-\sqrt{34}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+4\sqrt{34}}{2*2}=\frac{0+4\sqrt{34}}{4}
=\frac{4\sqrt{34}}{4}
=\sqrt{34}
$