1/2a=a+5

Solution for 1/2a=a+5 equation:

1/2a=a+5

We move all terms to the left:

1/2a-(a+5)=0


Domain of the equation: 2a!=0

a!=0/2

a!=0

a∈R


We get rid of parentheses

1/2a-a-5=0

We multiply all the terms by the denominator


-a*2a-5*2a+1=0

Wy multiply elements

-2a^2-10a+1=0

a = -2; b = -10; c = +1;
Δ = b2-4ac
Δ = -102-4·(-2)·1
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-6\sqrt{3}}{2*-2}=\frac{10-6\sqrt{3}}{-4}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+6\sqrt{3}}{2*-2}=\frac{10+6\sqrt{3}}{-4}
$