2a+7-1/2a=40+3a

Solution for 2a+7-1/2a=40+3a equation:

2a+7-1/2a=40+3a

We move all terms to the left:

2a+7-1/2a-(40+3a)=0


Domain of the equation: 2a!=0

a!=0/2

a!=0

a∈R


We add all the numbers together, and all the variables

2a-1/2a-(3a+40)+7=0

We get rid of parentheses

2a-1/2a-3a-40+7=0

We multiply all the terms by the denominator


2a*2a-3a*2a-40*2a+7*2a-1=0

Wy multiply elements

4a^2-6a^2-80a+14a-1=0

We add all the numbers together, and all the variables

-2a^2-66a-1=0

a = -2; b = -66; c = -1;
Δ = b2-4ac
Δ = -662-4·(-2)·(-1)
Δ = 4348
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:

$\sqrt{\Delta}=\sqrt{4348}=\sqrt{4*1087}=\sqrt{4}*\sqrt{1087}=2\sqrt{1087}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-66)-2\sqrt{1087}}{2*-2}=\frac{66-2\sqrt{1087}}{-4}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-66)+2\sqrt{1087}}{2*-2}=\frac{66+2\sqrt{1087}}{-4}
$