10k^2+5k-4=0

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Solution for 10k^2+5k-4=0 equation:


Simplifying
10k2 + 5k + -4 = 0

Reorder the terms:
-4 + 5k + 10k2 = 0

Solving
-4 + 5k + 10k2 = 0

Solving for variable 'k'.

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

Divide each side by '10'.
-0.4 + 0.5k + k2 = 0

Move the constant term to the right:

Add '0.4' to each side of the equation.
-0.4 + 0.5k + 0.4 + k2 = 0 + 0.4

Reorder the terms:
-0.4 + 0.4 + 0.5k + k2 = 0 + 0.4

Combine like terms: -0.4 + 0.4 = 0.0
0.0 + 0.5k + k2 = 0 + 0.4
0.5k + k2 = 0 + 0.4

Combine like terms: 0 + 0.4 = 0.4
0.5k + k2 = 0.4

The k term is 0.5k.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5k + 0.0625 + k2 = 0.4 + 0.0625

Reorder the terms:
0.0625 + 0.5k + k2 = 0.4 + 0.0625

Combine like terms: 0.4 + 0.0625 = 0.4625
0.0625 + 0.5k + k2 = 0.4625

Factor a perfect square on the left side:
(k + 0.25)(k + 0.25) = 0.4625

Calculate the square root of the right side: 0.680073525

Break this problem into two subproblems by setting 
(k + 0.25) equal to 0.680073525 and -0.680073525.

Subproblem 1

k + 0.25 = 0.680073525 Simplifying k + 0.25 = 0.680073525 Reorder the terms: 0.25 + k = 0.680073525 Solving 0.25 + k = 0.680073525 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + k = 0.680073525 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + k = 0.680073525 + -0.25 k = 0.680073525 + -0.25 Combine like terms: 0.680073525 + -0.25 = 0.430073525 k = 0.430073525 Simplifying k = 0.430073525

Subproblem 2

k + 0.25 = -0.680073525 Simplifying k + 0.25 = -0.680073525 Reorder the terms: 0.25 + k = -0.680073525 Solving 0.25 + k = -0.680073525 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + k = -0.680073525 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + k = -0.680073525 + -0.25 k = -0.680073525 + -0.25 Combine like terms: -0.680073525 + -0.25 = -0.930073525 k = -0.930073525 Simplifying k = -0.930073525

Solution

The solution to the problem is based on the solutions from the subproblems. k = {0.430073525, -0.930073525}

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