7k^2-5k-21=0

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Solution for 7k^2-5k-21=0 equation:


Simplifying
7k2 + -5k + -21 = 0

Reorder the terms:
-21 + -5k + 7k2 = 0

Solving
-21 + -5k + 7k2 = 0

Solving for variable 'k'.

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

Divide each side by '7'.
-3 + -0.7142857143k + k2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -0.7142857143k + 3 + k2 = 0 + 3

Reorder the terms:
-3 + 3 + -0.7142857143k + k2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -0.7142857143k + k2 = 0 + 3
-0.7142857143k + k2 = 0 + 3

Combine like terms: 0 + 3 = 3
-0.7142857143k + k2 = 3

The k term is -0.7142857143k.  Take half its coefficient (-0.3571428572).
Square it (0.1275510204) and add it to both sides.

Add '0.1275510204' to each side of the equation.
-0.7142857143k + 0.1275510204 + k2 = 3 + 0.1275510204

Reorder the terms:
0.1275510204 + -0.7142857143k + k2 = 3 + 0.1275510204

Combine like terms: 3 + 0.1275510204 = 3.1275510204
0.1275510204 + -0.7142857143k + k2 = 3.1275510204

Factor a perfect square on the left side:
(k + -0.3571428572)(k + -0.3571428572) = 3.1275510204

Calculate the square root of the right side: 1.768488343

Break this problem into two subproblems by setting 
(k + -0.3571428572) equal to 1.768488343 and -1.768488343.

Subproblem 1

k + -0.3571428572 = 1.768488343 Simplifying k + -0.3571428572 = 1.768488343 Reorder the terms: -0.3571428572 + k = 1.768488343 Solving -0.3571428572 + k = 1.768488343 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + k = 1.768488343 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + k = 1.768488343 + 0.3571428572 k = 1.768488343 + 0.3571428572 Combine like terms: 1.768488343 + 0.3571428572 = 2.1256312002 k = 2.1256312002 Simplifying k = 2.1256312002

Subproblem 2

k + -0.3571428572 = -1.768488343 Simplifying k + -0.3571428572 = -1.768488343 Reorder the terms: -0.3571428572 + k = -1.768488343 Solving -0.3571428572 + k = -1.768488343 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + k = -1.768488343 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + k = -1.768488343 + 0.3571428572 k = -1.768488343 + 0.3571428572 Combine like terms: -1.768488343 + 0.3571428572 = -1.4113454858 k = -1.4113454858 Simplifying k = -1.4113454858

Solution

The solution to the problem is based on the solutions from the subproblems. k = {2.1256312002, -1.4113454858}

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