-1/2x=4x+7

Solution for -1/2x=4x+7 equation:

-1/2x=4x+7

We move all terms to the left:

-1/2x-(4x+7)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

-1/2x-4x-7=0

We multiply all the terms by the denominator


-4x*2x-7*2x-1=0

Wy multiply elements

-8x^2-14x-1=0

a = -8; b = -14; c = -1;
Δ = b2-4ac
Δ = -142-4·(-8)·(-1)
Δ = 164
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{164}=\sqrt{4*41}=\sqrt{4}*\sqrt{41}=2\sqrt{41}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{41}}{2*-8}=\frac{14-2\sqrt{41}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{41}}{2*-8}=\frac{14+2\sqrt{41}}{-16}
$