4k^2-5k-1=0

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Solution for 4k^2-5k-1=0 equation:


Simplifying
4k2 + -5k + -1 = 0

Reorder the terms:
-1 + -5k + 4k2 = 0

Solving
-1 + -5k + 4k2 = 0

Solving for variable 'k'.

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

Divide each side by '4'.
-0.25 + -1.25k + k2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -1.25k + 0.25 + k2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -1.25k + k2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -1.25k + k2 = 0 + 0.25
-1.25k + k2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-1.25k + k2 = 0.25

The k term is -1.25k.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25k + 0.390625 + k2 = 0.25 + 0.390625

Reorder the terms:
0.390625 + -1.25k + k2 = 0.25 + 0.390625

Combine like terms: 0.25 + 0.390625 = 0.640625
0.390625 + -1.25k + k2 = 0.640625

Factor a perfect square on the left side:
(k + -0.625)(k + -0.625) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(k + -0.625) equal to 0.80039053 and -0.80039053.

Subproblem 1

k + -0.625 = 0.80039053 Simplifying k + -0.625 = 0.80039053 Reorder the terms: -0.625 + k = 0.80039053 Solving -0.625 + k = 0.80039053 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + k = 0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + k = 0.80039053 + 0.625 k = 0.80039053 + 0.625 Combine like terms: 0.80039053 + 0.625 = 1.42539053 k = 1.42539053 Simplifying k = 1.42539053

Subproblem 2

k + -0.625 = -0.80039053 Simplifying k + -0.625 = -0.80039053 Reorder the terms: -0.625 + k = -0.80039053 Solving -0.625 + k = -0.80039053 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + k = -0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + k = -0.80039053 + 0.625 k = -0.80039053 + 0.625 Combine like terms: -0.80039053 + 0.625 = -0.17539053 k = -0.17539053 Simplifying k = -0.17539053

Solution

The solution to the problem is based on the solutions from the subproblems. k = {1.42539053, -0.17539053}

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