31=8w+3w(7-w)

Solution for 31=8w+3w(7-w) equation:

31=8w+3w(7-w)

We move all terms to the left:

31-(8w+3w(7-w))=0

We add all the numbers together, and all the variables

-(8w+3w(-1w+7))+31=0


We calculate terms in parentheses: -(8w+3w(-1w+7)), so:

8w+3w(-1w+7)

We multiply parentheses

-3w^2+8w+21w

We add all the numbers together, and all the variables

-3w^2+29w

Back to the equation:

-(-3w^2+29w)


We get rid of parentheses

3w^2-29w+31=0

a = 3; b = -29; c = +31;
Δ = b2-4ac
Δ = -292-4·3·31
Δ = 469
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$w_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$w_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$w_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-29)-\sqrt{469}}{2*3}=\frac{29-\sqrt{469}}{6}
$
$w_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-29)+\sqrt{469}}{2*3}=\frac{29+\sqrt{469}}{6}
$