(1)/(2)a+7=2a-2

Solution for (1)/(2)a+7=2a-2 equation:

(1)/(2)a+7=2a-2

We move all terms to the left:

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


Domain of the equation: 2a!=0

a!=0/2

a!=0

a∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-4a^2+18a+1=0

a = -4; b = 18; c = +1;
Δ = b2-4ac
Δ = 182-4·(-4)·1
Δ = 340
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:

$\sqrt{\Delta}=\sqrt{340}=\sqrt{4*85}=\sqrt{4}*\sqrt{85}=2\sqrt{85}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-2\sqrt{85}}{2*-4}=\frac{-18-2\sqrt{85}}{-8}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+2\sqrt{85}}{2*-4}=\frac{-18+2\sqrt{85}}{-8}
$