1/4a-5=8a+7

Solution for 1/4a-5=8a+7 equation:

1/4a-5=8a+7

We move all terms to the left:

1/4a-5-(8a+7)=0


Domain of the equation: 4a!=0

a!=0/4

a!=0

a∈R


We get rid of parentheses

1/4a-8a-7-5=0

We multiply all the terms by the denominator


-8a*4a-7*4a-5*4a+1=0

Wy multiply elements

-32a^2-28a-20a+1=0

We add all the numbers together, and all the variables

-32a^2-48a+1=0

a = -32; b = -48; c = +1;
Δ = b2-4ac
Δ = -482-4·(-32)·1
Δ = 2432
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:

$\sqrt{\Delta}=\sqrt{2432}=\sqrt{64*38}=\sqrt{64}*\sqrt{38}=8\sqrt{38}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-48)-8\sqrt{38}}{2*-32}=\frac{48-8\sqrt{38}}{-64}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-48)+8\sqrt{38}}{2*-32}=\frac{48+8\sqrt{38}}{-64}
$