10r^2-9r+1=0

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Solution for 10r^2-9r+1=0 equation:


Simplifying
10r2 + -9r + 1 = 0

Reorder the terms:
1 + -9r + 10r2 = 0

Solving
1 + -9r + 10r2 = 0

Solving for variable 'r'.

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

Divide each side by '10'.
0.1 + -0.9r + r2 = 0

Move the constant term to the right:

Add '-0.1' to each side of the equation.
0.1 + -0.9r + -0.1 + r2 = 0 + -0.1

Reorder the terms:
0.1 + -0.1 + -0.9r + r2 = 0 + -0.1

Combine like terms: 0.1 + -0.1 = 0.0
0.0 + -0.9r + r2 = 0 + -0.1
-0.9r + r2 = 0 + -0.1

Combine like terms: 0 + -0.1 = -0.1
-0.9r + r2 = -0.1

The r term is -0.9r.  Take half its coefficient (-0.45).
Square it (0.2025) and add it to both sides.

Add '0.2025' to each side of the equation.
-0.9r + 0.2025 + r2 = -0.1 + 0.2025

Reorder the terms:
0.2025 + -0.9r + r2 = -0.1 + 0.2025

Combine like terms: -0.1 + 0.2025 = 0.1025
0.2025 + -0.9r + r2 = 0.1025

Factor a perfect square on the left side:
(r + -0.45)(r + -0.45) = 0.1025

Calculate the square root of the right side: 0.320156212

Break this problem into two subproblems by setting 
(r + -0.45) equal to 0.320156212 and -0.320156212.

Subproblem 1

r + -0.45 = 0.320156212 Simplifying r + -0.45 = 0.320156212 Reorder the terms: -0.45 + r = 0.320156212 Solving -0.45 + r = 0.320156212 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + r = 0.320156212 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + r = 0.320156212 + 0.45 r = 0.320156212 + 0.45 Combine like terms: 0.320156212 + 0.45 = 0.770156212 r = 0.770156212 Simplifying r = 0.770156212

Subproblem 2

r + -0.45 = -0.320156212 Simplifying r + -0.45 = -0.320156212 Reorder the terms: -0.45 + r = -0.320156212 Solving -0.45 + r = -0.320156212 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + r = -0.320156212 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + r = -0.320156212 + 0.45 r = -0.320156212 + 0.45 Combine like terms: -0.320156212 + 0.45 = 0.129843788 r = 0.129843788 Simplifying r = 0.129843788

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.770156212, 0.129843788}

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