1/4x=3x-11

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Solution for 1/4x=3x-11 equation:



1/4x=3x-11
We move all terms to the left:
1/4x-(3x-11)=0
Domain of the equation: 4x!=0
x!=0/4
x!=0
x∈R
We get rid of parentheses
1/4x-3x+11=0
We multiply all the terms by the denominator
-3x*4x+11*4x+1=0
Wy multiply elements
-12x^2+44x+1=0
a = -12; b = 44; c = +1;
Δ = b2-4ac
Δ = 442-4·(-12)·1
Δ = 1984
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:
$\sqrt{\Delta}=\sqrt{1984}=\sqrt{64*31}=\sqrt{64}*\sqrt{31}=8\sqrt{31}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(44)-8\sqrt{31}}{2*-12}=\frac{-44-8\sqrt{31}}{-24} $
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(44)+8\sqrt{31}}{2*-12}=\frac{-44+8\sqrt{31}}{-24} $

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