3-1/2x=5x+18

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Solution for 3-1/2x=5x+18 equation:



3-1/2x=5x+18
We move all terms to the left:
3-1/2x-(5x+18)=0
Domain of the equation: 2x!=0
x!=0/2
x!=0
x∈R
We get rid of parentheses
-1/2x-5x-18+3=0
We multiply all the terms by the denominator
-5x*2x-18*2x+3*2x-1=0
Wy multiply elements
-10x^2-36x+6x-1=0
We add all the numbers together, and all the variables
-10x^2-30x-1=0
a = -10; b = -30; c = -1;
Δ = b2-4ac
Δ = -302-4·(-10)·(-1)
Δ = 860
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:
$\sqrt{\Delta}=\sqrt{860}=\sqrt{4*215}=\sqrt{4}*\sqrt{215}=2\sqrt{215}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{215}}{2*-10}=\frac{30-2\sqrt{215}}{-20} $
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{215}}{2*-10}=\frac{30+2\sqrt{215}}{-20} $

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