3x(5+2x)-5(3x+7)=1/2

Solution for 3x(5+2x)-5(3x+7)=1/2 equation:

3x(5+2x)-5(3x+7)=1/2

We move all terms to the left:

3x(5+2x)-5(3x+7)-(1/2)=0

We add all the numbers together, and all the variables

3x(2x+5)-5(3x+7)-(+1/2)=0

We multiply parentheses

6x^2+15x-15x-35-(+1/2)=0

We get rid of parentheses

6x^2+15x-15x-35-1/2=0

We multiply all the terms by the denominator


6x^2*2+15x*2-15x*2-1-35*2=0

We add all the numbers together, and all the variables

6x^2*2+15x*2-15x*2-71=0

Wy multiply elements

12x^2+30x-30x-71=0

We add all the numbers together, and all the variables

12x^2-71=0

a = 12; b = 0; c = -71;
Δ = b2-4ac
Δ = 02-4·12·(-71)
Δ = 3408
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:

$\sqrt{\Delta}=\sqrt{3408}=\sqrt{16*213}=\sqrt{16}*\sqrt{213}=4\sqrt{213}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-4\sqrt{213}}{2*12}=\frac{0-4\sqrt{213}}{24}
=-\frac{4\sqrt{213}}{24}
=-\frac{\sqrt{213}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+4\sqrt{213}}{2*12}=\frac{0+4\sqrt{213}}{24}
=\frac{4\sqrt{213}}{24}
=\frac{\sqrt{213}}{6}
$