0.5k(k+1)+K+1=0.5(k+1)(k+2)

Solution for 0.5k(k+1)+K+1=0.5(k+1)(k+2) equation:

0.5k(k+1)++1=0.5(k+1)(k+2)

We move all terms to the left:

0.5k(k+1)++1-(0.5(k+1)(k+2))=0

We add all the numbers together, and all the variables

0.5k(k+1)-(0.5(k+1)(k+2))=0

We multiply parentheses

0k^2+0k-(0.5(k+1)(k+2))=0

We multiply parentheses ..

0k^2-(0.5(+k^2+2k+k+2))+0k=0


We calculate terms in parentheses: -(0.5(+k^2+2k+k+2)), so:

0.5(+k^2+2k+k+2)

We multiply parentheses

0.5k^2+k+0.5k+1

We add all the numbers together, and all the variables

0.5k^2+1.5k+1

Back to the equation:

-(0.5k^2+1.5k+1)


We add all the numbers together, and all the variables

k^2+k-(0.5k^2+1.5k+1)=0

We get rid of parentheses

k^2-0.5k^2+k-1.5k-1=0

We add all the numbers together, and all the variables

0.5k^2-0.5k-1=0

a = 0.5; b = -0.5; c = -1;
Δ = b2-4ac
Δ = -0.52-4·0.5·(-1)
Δ = 2.25
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-0.5)-\sqrt{2.25}}{2*0.5}=\frac{0.5-\sqrt{2.25}}{1}
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-0.5)+\sqrt{2.25}}{2*0.5}=\frac{0.5+\sqrt{2.25}}{1}
$