1/2t-3=2t+3

Solution for 1/2t-3=2t+3 equation:

1/2t-3=2t+3

We move all terms to the left:

1/2t-3-(2t+3)=0


Domain of the equation: 2t!=0

t!=0/2

t!=0

t∈R


We get rid of parentheses

1/2t-2t-3-3=0

We multiply all the terms by the denominator


-2t*2t-3*2t-3*2t+1=0

Wy multiply elements

-4t^2-6t-6t+1=0

We add all the numbers together, and all the variables

-4t^2-12t+1=0

a = -4; b = -12; c = +1;
Δ = b2-4ac
Δ = -122-4·(-4)·1
Δ = 160
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{160}=\sqrt{16*10}=\sqrt{16}*\sqrt{10}=4\sqrt{10}$
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-4\sqrt{10}}{2*-4}=\frac{12-4\sqrt{10}}{-8}
$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+4\sqrt{10}}{2*-4}=\frac{12+4\sqrt{10}}{-8}
$