1/4x-5=10-x

Solution for 1/4x-5=10-x equation:

1/4x-5=10-x

We move all terms to the left:

1/4x-5-(10-x)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We add all the numbers together, and all the variables

1/4x-(-1x+10)-5=0

We get rid of parentheses

1/4x+1x-10-5=0

We multiply all the terms by the denominator


1x*4x-10*4x-5*4x+1=0

Wy multiply elements

4x^2-40x-20x+1=0

We add all the numbers together, and all the variables

4x^2-60x+1=0

a = 4; b = -60; c = +1;
Δ = b2-4ac
Δ = -602-4·4·1
Δ = 3584
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:

$\sqrt{\Delta}=\sqrt{3584}=\sqrt{256*14}=\sqrt{256}*\sqrt{14}=16\sqrt{14}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-60)-16\sqrt{14}}{2*4}=\frac{60-16\sqrt{14}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-60)+16\sqrt{14}}{2*4}=\frac{60+16\sqrt{14}}{8}
$