31/2a-51/4a+3/4a=1

Solution for 31/2a-51/4a+3/4a=1 equation:

31/2a-51/4a+3/4a=1

We move all terms to the left:

31/2a-51/4a+3/4a-(1)=0


Domain of the equation: 2a!=0

a!=0/2

a!=0

a∈R



Domain of the equation: 4a!=0

a!=0/4

a!=0

a∈R


We calculate fractions


124a/8a^2+(6a-51)/8a^2-1=0

We multiply all the terms by the denominator


124a+(6a-51)-1*8a^2=0

Wy multiply elements

-8a^2+124a+(6a-51)=0

We get rid of parentheses

-8a^2+124a+6a-51=0

We add all the numbers together, and all the variables

-8a^2+130a-51=0

a = -8; b = 130; c = -51;
Δ = b2-4ac
Δ = 1302-4·(-8)·(-51)
Δ = 15268
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:

$\sqrt{\Delta}=\sqrt{15268}=\sqrt{4*3817}=\sqrt{4}*\sqrt{3817}=2\sqrt{3817}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(130)-2\sqrt{3817}}{2*-8}=\frac{-130-2\sqrt{3817}}{-16}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(130)+2\sqrt{3817}}{2*-8}=\frac{-130+2\sqrt{3817}}{-16}
$