1/2*7q+2=4q-5

Solution for 1/2*7q+2=4q-5 equation:

1/2*7q+2=4q-5

We move all terms to the left:

1/2*7q+2-(4q-5)=0


Domain of the equation: 2*7q!=0

q!=0/1

q!=0

q∈R


We get rid of parentheses

1/2*7q-4q+5+2=0

We multiply all the terms by the denominator


-4q*2*7q+5*2*7q+2*2*7q+1=0

Wy multiply elements

-56q^2*7+70q*7+28q*7+1=0

Wy multiply elements

-392q^2+490q+196q+1=0

We add all the numbers together, and all the variables

-392q^2+686q+1=0

a = -392; b = 686; c = +1;
Δ = b2-4ac
Δ = 6862-4·(-392)·1
Δ = 472164
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{472164}=\sqrt{196*2409}=\sqrt{196}*\sqrt{2409}=14\sqrt{2409}$
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(686)-14\sqrt{2409}}{2*-392}=\frac{-686-14\sqrt{2409}}{-784}
$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(686)+14\sqrt{2409}}{2*-392}=\frac{-686+14\sqrt{2409}}{-784}
$