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1/2x+5=3x-10
We move all terms to the left:
1/2x+5-(3x-10)=0
Domain of the equation: 2x!=0We get rid of parentheses
x!=0/2
x!=0
x∈R
1/2x-3x+10+5=0
We multiply all the terms by the denominator
-3x*2x+10*2x+5*2x+1=0
Wy multiply elements
-6x^2+20x+10x+1=0
We add all the numbers together, and all the variables
-6x^2+30x+1=0
a = -6; b = 30; c = +1;
Δ = b2-4ac
Δ = 302-4·(-6)·1
Δ = 924
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}$$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$
The end solution:
$\sqrt{\Delta}=\sqrt{924}=\sqrt{4*231}=\sqrt{4}*\sqrt{231}=2\sqrt{231}$$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(30)-2\sqrt{231}}{2*-6}=\frac{-30-2\sqrt{231}}{-12} $$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(30)+2\sqrt{231}}{2*-6}=\frac{-30+2\sqrt{231}}{-12} $
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