31=(-1/2)(8x+22)+2

Solution for 31=(-1/2)(8x+22)+2 equation:

31=(-1/2)(8x+22)+2

We move all terms to the left:

31-((-1/2)(8x+22)+2)=0


Domain of the equation: 2)(8x+22)+2)!=0

x∈R


We multiply parentheses ..

-((-8x^2-1/2*22)+2)+31=0

We multiply all the terms by the denominator


-((-8x^2-1+31*2*22)+2)=0


We calculate terms in parentheses: -((-8x^2-1+31*2*22)+2), so:

(-8x^2-1+31*2*22)+2

We get rid of parentheses

-8x^2-1+2+31*2*22

We add all the numbers together, and all the variables

-8x^2+1365

Back to the equation:

-(-8x^2+1365)


We get rid of parentheses

8x^2-1365=0

a = 8; b = 0; c = -1365;
Δ = b2-4ac
Δ = 02-4·8·(-1365)
Δ = 43680
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:

$\sqrt{\Delta}=\sqrt{43680}=\sqrt{16*2730}=\sqrt{16}*\sqrt{2730}=4\sqrt{2730}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-4\sqrt{2730}}{2*8}=\frac{0-4\sqrt{2730}}{16}
=-\frac{4\sqrt{2730}}{16}
=-\frac{\sqrt{2730}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+4\sqrt{2730}}{2*8}=\frac{0+4\sqrt{2730}}{16}
=\frac{4\sqrt{2730}}{16}
=\frac{\sqrt{2730}}{4}
$