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 = -10
-21r + 10r2 = -10

Solving
-21r + 10r2 = -10

Solving for variable 'r'.

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

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

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

Divide each side by '10'.
1 + -2.1r + r2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -2.1r + -1 + r2 = 0 + -1

Reorder the terms:
1 + -1 + -2.1r + r2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -2.1r + r2 = 0 + -1
-2.1r + r2 = 0 + -1

Combine like terms: 0 + -1 = -1
-2.1r + r2 = -1

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 = -1 + 1.1025

Reorder the terms:
1.1025 + -2.1r + r2 = -1 + 1.1025

Combine like terms: -1 + 1.1025 = 0.1025
1.1025 + -2.1r + r2 = 0.1025

Factor a perfect square on the left side:
(r + -1.05)(r + -1.05) = 0.1025

Calculate the square root of the right side: 0.320156212

Break this problem into two subproblems by setting 
(r + -1.05) equal to 0.320156212 and -0.320156212.

Subproblem 1

r + -1.05 = 0.320156212 Simplifying r + -1.05 = 0.320156212 Reorder the terms: -1.05 + r = 0.320156212 Solving -1.05 + r = 0.320156212 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 = 0.320156212 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + r = 0.320156212 + 1.05 r = 0.320156212 + 1.05 Combine like terms: 0.320156212 + 1.05 = 1.370156212 r = 1.370156212 Simplifying r = 1.370156212

Subproblem 2

r + -1.05 = -0.320156212 Simplifying r + -1.05 = -0.320156212 Reorder the terms: -1.05 + r = -0.320156212 Solving -1.05 + r = -0.320156212 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 = -0.320156212 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + r = -0.320156212 + 1.05 r = -0.320156212 + 1.05 Combine like terms: -0.320156212 + 1.05 = 0.729843788 r = 0.729843788 Simplifying r = 0.729843788

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.370156212, 0.729843788}

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