4k(k+1)=25

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Solution for 4k(k+1)=25 equation:


Simplifying
4k(k + 1) = 25

Reorder the terms:
4k(1 + k) = 25
(1 * 4k + k * 4k) = 25
(4k + 4k2) = 25

Solving
4k + 4k2 = 25

Solving for variable 'k'.

Reorder the terms:
-25 + 4k + 4k2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + 4k + 4k2 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-6.25 + k + k2 = 0

Move the constant term to the right:

Add '6.25' to each side of the equation.
-6.25 + k + 6.25 + k2 = 0 + 6.25

Reorder the terms:
-6.25 + 6.25 + k + k2 = 0 + 6.25

Combine like terms: -6.25 + 6.25 = 0.00
0.00 + k + k2 = 0 + 6.25
k + k2 = 0 + 6.25

Combine like terms: 0 + 6.25 = 6.25
k + k2 = 6.25

The k term is k.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + k2 = 6.25 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + k2 = 6.25 + 0.25

Combine like terms: 6.25 + 0.25 = 6.5
1.25 + k2 = 6.5

Factor a perfect square on the left side:
(k + 0.5)(k + 0.5) = 6.5

Calculate the square root of the right side: 2.549509757

Break this problem into two subproblems by setting 
(k + 0.5) equal to 2.549509757 and -2.549509757.

Subproblem 1

k + 0.5 = 2.549509757 Simplifying k + 0.5 = 2.549509757 Reorder the terms: 0.5 + k = 2.549509757 Solving 0.5 + k = 2.549509757 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = 2.549509757 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 2.549509757 + -0.5 k = 2.549509757 + -0.5 Combine like terms: 2.549509757 + -0.5 = 2.049509757 k = 2.049509757 Simplifying k = 2.049509757

Subproblem 2

k + 0.5 = -2.549509757 Simplifying k + 0.5 = -2.549509757 Reorder the terms: 0.5 + k = -2.549509757 Solving 0.5 + k = -2.549509757 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = -2.549509757 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -2.549509757 + -0.5 k = -2.549509757 + -0.5 Combine like terms: -2.549509757 + -0.5 = -3.049509757 k = -3.049509757 Simplifying k = -3.049509757

Solution

The solution to the problem is based on the solutions from the subproblems. k = {2.049509757, -3.049509757}

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