14r^2-21r-6=0

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Solution for 14r^2-21r-6=0 equation:


Simplifying
14r2 + -21r + -6 = 0

Reorder the terms:
-6 + -21r + 14r2 = 0

Solving
-6 + -21r + 14r2 = 0

Solving for variable 'r'.

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

Divide each side by '14'.
-0.4285714286 + -1.5r + r2 = 0

Move the constant term to the right:

Add '0.4285714286' to each side of the equation.
-0.4285714286 + -1.5r + 0.4285714286 + r2 = 0 + 0.4285714286

Reorder the terms:
-0.4285714286 + 0.4285714286 + -1.5r + r2 = 0 + 0.4285714286

Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000
0.0000000000 + -1.5r + r2 = 0 + 0.4285714286
-1.5r + r2 = 0 + 0.4285714286

Combine like terms: 0 + 0.4285714286 = 0.4285714286
-1.5r + r2 = 0.4285714286

The r term is -1.5r.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5r + 0.5625 + r2 = 0.4285714286 + 0.5625

Reorder the terms:
0.5625 + -1.5r + r2 = 0.4285714286 + 0.5625

Combine like terms: 0.4285714286 + 0.5625 = 0.9910714286
0.5625 + -1.5r + r2 = 0.9910714286

Factor a perfect square on the left side:
(r + -0.75)(r + -0.75) = 0.9910714286

Calculate the square root of the right side: 0.995525705

Break this problem into two subproblems by setting 
(r + -0.75) equal to 0.995525705 and -0.995525705.

Subproblem 1

r + -0.75 = 0.995525705 Simplifying r + -0.75 = 0.995525705 Reorder the terms: -0.75 + r = 0.995525705 Solving -0.75 + r = 0.995525705 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = 0.995525705 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = 0.995525705 + 0.75 r = 0.995525705 + 0.75 Combine like terms: 0.995525705 + 0.75 = 1.745525705 r = 1.745525705 Simplifying r = 1.745525705

Subproblem 2

r + -0.75 = -0.995525705 Simplifying r + -0.75 = -0.995525705 Reorder the terms: -0.75 + r = -0.995525705 Solving -0.75 + r = -0.995525705 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = -0.995525705 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = -0.995525705 + 0.75 r = -0.995525705 + 0.75 Combine like terms: -0.995525705 + 0.75 = -0.245525705 r = -0.245525705 Simplifying r = -0.245525705

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.745525705, -0.245525705}

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