5=3t+1/2*9.8t^2

Solution for 5=3t+1/2*9.8t^2 equation:

5=3t+1/2*9.8t^2

We move all terms to the left:

5-(3t+1/2*9.8t^2)=0


Domain of the equation: 2*9.8t^2)!=0

t!=0/1

t!=0

t∈R


We get rid of parentheses

-3t-1/2*9.8t^2+5=0

We multiply all the terms by the denominator


-3t*2*9.8t^2+5*2*9.8t^2-1=0

Wy multiply elements

-54t^2*9+90t*9-1=0

Wy multiply elements

-486t^2+810t-1=0

a = -486; b = 810; c = -1;
Δ = b2-4ac
Δ = 8102-4·(-486)·(-1)
Δ = 654156
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:

$\sqrt{\Delta}=\sqrt{654156}=\sqrt{324*2019}=\sqrt{324}*\sqrt{2019}=18\sqrt{2019}$
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(810)-18\sqrt{2019}}{2*-486}=\frac{-810-18\sqrt{2019}}{-972}
$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(810)+18\sqrt{2019}}{2*-486}=\frac{-810+18\sqrt{2019}}{-972}
$