1/2a=3a-7

Solution for 1/2a=3a-7 equation:

1/2a=3a-7

We move all terms to the left:

1/2a-(3a-7)=0


Domain of the equation: 2a!=0

a!=0/2

a!=0

a∈R


We get rid of parentheses

1/2a-3a+7=0

We multiply all the terms by the denominator


-3a*2a+7*2a+1=0

Wy multiply elements

-6a^2+14a+1=0

a = -6; b = 14; c = +1;
Δ = b2-4ac
Δ = 142-4·(-6)·1
Δ = 220
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{220}=\sqrt{4*55}=\sqrt{4}*\sqrt{55}=2\sqrt{55}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(14)-2\sqrt{55}}{2*-6}=\frac{-14-2\sqrt{55}}{-12}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(14)+2\sqrt{55}}{2*-6}=\frac{-14+2\sqrt{55}}{-12}
$