1/2x+7=4x-21

Solution for 1/2x+7=4x-21 equation:

1/2x+7=4x-21

We move all terms to the left:

1/2x+7-(4x-21)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-4x+21+7=0

We multiply all the terms by the denominator


-4x*2x+21*2x+7*2x+1=0

Wy multiply elements

-8x^2+42x+14x+1=0

We add all the numbers together, and all the variables

-8x^2+56x+1=0

a = -8; b = 56; c = +1;
Δ = b2-4ac
Δ = 562-4·(-8)·1
Δ = 3168
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:

$\sqrt{\Delta}=\sqrt{3168}=\sqrt{144*22}=\sqrt{144}*\sqrt{22}=12\sqrt{22}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(56)-12\sqrt{22}}{2*-8}=\frac{-56-12\sqrt{22}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(56)+12\sqrt{22}}{2*-8}=\frac{-56+12\sqrt{22}}{-16}
$