1/2a-9=4a+5

Solution for 1/2a-9=4a+5 equation:

1/2a-9=4a+5

We move all terms to the left:

1/2a-9-(4a+5)=0


Domain of the equation: 2a!=0

a!=0/2

a!=0

a∈R


We get rid of parentheses

1/2a-4a-5-9=0

We multiply all the terms by the denominator


-4a*2a-5*2a-9*2a+1=0

Wy multiply elements

-8a^2-10a-18a+1=0

We add all the numbers together, and all the variables

-8a^2-28a+1=0

a = -8; b = -28; c = +1;
Δ = b2-4ac
Δ = -282-4·(-8)·1
Δ = 816
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:

$\sqrt{\Delta}=\sqrt{816}=\sqrt{16*51}=\sqrt{16}*\sqrt{51}=4\sqrt{51}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-28)-4\sqrt{51}}{2*-8}=\frac{28-4\sqrt{51}}{-16}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-28)+4\sqrt{51}}{2*-8}=\frac{28+4\sqrt{51}}{-16}
$