10r^2-21r=-4+6

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Solution for 10r^2-21r=-4+6 equation:


Simplifying
10r2 + -21r = -4 + 6

Reorder the terms:
-21r + 10r2 = -4 + 6

Combine like terms: -4 + 6 = 2
-21r + 10r2 = 2

Solving
-21r + 10r2 = 2

Solving for variable 'r'.

Reorder the terms:
-2 + -21r + 10r2 = 2 + -2

Combine like terms: 2 + -2 = 0
-2 + -21r + 10r2 = 0

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

Divide each side by '10'.
-0.2 + -2.1r + r2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -2.1r + 0.2 + r2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -2.1r + r2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -2.1r + r2 = 0 + 0.2
-2.1r + r2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-2.1r + r2 = 0.2

The r term is -2.1r.  Take half its coefficient (-1.05).
Square it (1.1025) and add it to both sides.

Add '1.1025' to each side of the equation.
-2.1r + 1.1025 + r2 = 0.2 + 1.1025

Reorder the terms:
1.1025 + -2.1r + r2 = 0.2 + 1.1025

Combine like terms: 0.2 + 1.1025 = 1.3025
1.1025 + -2.1r + r2 = 1.3025

Factor a perfect square on the left side:
(r + -1.05)(r + -1.05) = 1.3025

Calculate the square root of the right side: 1.141271221

Break this problem into two subproblems by setting 
(r + -1.05) equal to 1.141271221 and -1.141271221.

Subproblem 1

r + -1.05 = 1.141271221 Simplifying r + -1.05 = 1.141271221 Reorder the terms: -1.05 + r = 1.141271221 Solving -1.05 + r = 1.141271221 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + r = 1.141271221 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + r = 1.141271221 + 1.05 r = 1.141271221 + 1.05 Combine like terms: 1.141271221 + 1.05 = 2.191271221 r = 2.191271221 Simplifying r = 2.191271221

Subproblem 2

r + -1.05 = -1.141271221 Simplifying r + -1.05 = -1.141271221 Reorder the terms: -1.05 + r = -1.141271221 Solving -1.05 + r = -1.141271221 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + r = -1.141271221 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + r = -1.141271221 + 1.05 r = -1.141271221 + 1.05 Combine like terms: -1.141271221 + 1.05 = -0.091271221 r = -0.091271221 Simplifying r = -0.091271221

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.191271221, -0.091271221}

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