1/a-4=a1+7

Solution for 1/a-4=a1+7 equation:

1/a-4=a1+7

We move all terms to the left:

1/a-4-(a1+7)=0


Domain of the equation: a!=0

a∈R


We add all the numbers together, and all the variables

1/a-(a+7)-4=0

We get rid of parentheses

1/a-a-7-4=0

We multiply all the terms by the denominator


-a*a-7*a-4*a+1=0

We add all the numbers together, and all the variables

-11a-a*a+1=0

Wy multiply elements

-1a^2-11a+1=0

a = -1; b = -11; c = +1;
Δ = b2-4ac
Δ = -112-4·(-1)·1
Δ = 125
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:

$\sqrt{\Delta}=\sqrt{125}=\sqrt{25*5}=\sqrt{25}*\sqrt{5}=5\sqrt{5}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-11)-5\sqrt{5}}{2*-1}=\frac{11-5\sqrt{5}}{-2}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-11)+5\sqrt{5}}{2*-1}=\frac{11+5\sqrt{5}}{-2}
$