21r^2-41r-42=0

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Solution for 21r^2-41r-42=0 equation:


Simplifying
21r2 + -41r + -42 = 0

Reorder the terms:
-42 + -41r + 21r2 = 0

Solving
-42 + -41r + 21r2 = 0

Solving for variable 'r'.

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

Divide each side by '21'.
-2 + -1.952380952r + r2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1.952380952r + 2 + r2 = 0 + 2

Reorder the terms:
-2 + 2 + -1.952380952r + r2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1.952380952r + r2 = 0 + 2
-1.952380952r + r2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1.952380952r + r2 = 2

The r term is -1.952380952r.  Take half its coefficient (-0.976190476).
Square it (0.9529478454) and add it to both sides.

Add '0.9529478454' to each side of the equation.
-1.952380952r + 0.9529478454 + r2 = 2 + 0.9529478454

Reorder the terms:
0.9529478454 + -1.952380952r + r2 = 2 + 0.9529478454

Combine like terms: 2 + 0.9529478454 = 2.9529478454
0.9529478454 + -1.952380952r + r2 = 2.9529478454

Factor a perfect square on the left side:
(r + -0.976190476)(r + -0.976190476) = 2.9529478454

Calculate the square root of the right side: 1.71841434

Break this problem into two subproblems by setting 
(r + -0.976190476) equal to 1.71841434 and -1.71841434.

Subproblem 1

r + -0.976190476 = 1.71841434 Simplifying r + -0.976190476 = 1.71841434 Reorder the terms: -0.976190476 + r = 1.71841434 Solving -0.976190476 + r = 1.71841434 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.976190476' to each side of the equation. -0.976190476 + 0.976190476 + r = 1.71841434 + 0.976190476 Combine like terms: -0.976190476 + 0.976190476 = 0.000000000 0.000000000 + r = 1.71841434 + 0.976190476 r = 1.71841434 + 0.976190476 Combine like terms: 1.71841434 + 0.976190476 = 2.694604816 r = 2.694604816 Simplifying r = 2.694604816

Subproblem 2

r + -0.976190476 = -1.71841434 Simplifying r + -0.976190476 = -1.71841434 Reorder the terms: -0.976190476 + r = -1.71841434 Solving -0.976190476 + r = -1.71841434 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.976190476' to each side of the equation. -0.976190476 + 0.976190476 + r = -1.71841434 + 0.976190476 Combine like terms: -0.976190476 + 0.976190476 = 0.000000000 0.000000000 + r = -1.71841434 + 0.976190476 r = -1.71841434 + 0.976190476 Combine like terms: -1.71841434 + 0.976190476 = -0.742223864 r = -0.742223864 Simplifying r = -0.742223864

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.694604816, -0.742223864}

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