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1/2x+1=2x-3
We move all terms to the left:
1/2x+1-(2x-3)=0
Domain of the equation: 2x!=0We get rid of parentheses
x!=0/2
x!=0
x∈R
1/2x-2x+3+1=0
We multiply all the terms by the denominator
-2x*2x+3*2x+1*2x+1=0
Wy multiply elements
-4x^2+6x+2x+1=0
We add all the numbers together, and all the variables
-4x^2+8x+1=0
a = -4; b = 8; c = +1;
Δ = b2-4ac
Δ = 82-4·(-4)·1
Δ = 80
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}$$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$
The end solution:
$\sqrt{\Delta}=\sqrt{80}=\sqrt{16*5}=\sqrt{16}*\sqrt{5}=4\sqrt{5}$$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-4\sqrt{5}}{2*-4}=\frac{-8-4\sqrt{5}}{-8} $$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+4\sqrt{5}}{2*-4}=\frac{-8+4\sqrt{5}}{-8} $
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