1/2x+7=2.5x-12

Solution for 1/2x+7=2.5x-12 equation:

1/2x+7=2.5x-12

We move all terms to the left:

1/2x+7-(2.5x-12)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-2.5x+12+7=0

We multiply all the terms by the denominator


-(2.5x)*2x+12*2x+7*2x+1=0

We add all the numbers together, and all the variables

-(+2.5x)*2x+12*2x+7*2x+1=0

We multiply parentheses

-4x^2+12*2x+7*2x+1=0

Wy multiply elements

-4x^2+24x+14x+1=0

We add all the numbers together, and all the variables

-4x^2+38x+1=0

a = -4; b = 38; c = +1;
Δ = b2-4ac
Δ = 382-4·(-4)·1
Δ = 1460
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:

$\sqrt{\Delta}=\sqrt{1460}=\sqrt{4*365}=\sqrt{4}*\sqrt{365}=2\sqrt{365}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(38)-2\sqrt{365}}{2*-4}=\frac{-38-2\sqrt{365}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(38)+2\sqrt{365}}{2*-4}=\frac{-38+2\sqrt{365}}{-8}
$