7r^2-9r+1=0

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Solution for 7r^2-9r+1=0 equation:


Simplifying
7r2 + -9r + 1 = 0

Reorder the terms:
1 + -9r + 7r2 = 0

Solving
1 + -9r + 7r2 = 0

Solving for variable 'r'.

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

Divide each side by '7'.
0.1428571429 + -1.285714286r + r2 = 0

Move the constant term to the right:

Add '-0.1428571429' to each side of the equation.
0.1428571429 + -1.285714286r + -0.1428571429 + r2 = 0 + -0.1428571429

Reorder the terms:
0.1428571429 + -0.1428571429 + -1.285714286r + r2 = 0 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + -1.285714286r + r2 = 0 + -0.1428571429
-1.285714286r + r2 = 0 + -0.1428571429

Combine like terms: 0 + -0.1428571429 = -0.1428571429
-1.285714286r + r2 = -0.1428571429

The r term is -1.285714286r.  Take half its coefficient (-0.642857143).
Square it (0.4132653063) and add it to both sides.

Add '0.4132653063' to each side of the equation.
-1.285714286r + 0.4132653063 + r2 = -0.1428571429 + 0.4132653063

Reorder the terms:
0.4132653063 + -1.285714286r + r2 = -0.1428571429 + 0.4132653063

Combine like terms: -0.1428571429 + 0.4132653063 = 0.2704081634
0.4132653063 + -1.285714286r + r2 = 0.2704081634

Factor a perfect square on the left side:
(r + -0.642857143)(r + -0.642857143) = 0.2704081634

Calculate the square root of the right side: 0.520007849

Break this problem into two subproblems by setting 
(r + -0.642857143) equal to 0.520007849 and -0.520007849.

Subproblem 1

r + -0.642857143 = 0.520007849 Simplifying r + -0.642857143 = 0.520007849 Reorder the terms: -0.642857143 + r = 0.520007849 Solving -0.642857143 + r = 0.520007849 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.642857143' to each side of the equation. -0.642857143 + 0.642857143 + r = 0.520007849 + 0.642857143 Combine like terms: -0.642857143 + 0.642857143 = 0.000000000 0.000000000 + r = 0.520007849 + 0.642857143 r = 0.520007849 + 0.642857143 Combine like terms: 0.520007849 + 0.642857143 = 1.162864992 r = 1.162864992 Simplifying r = 1.162864992

Subproblem 2

r + -0.642857143 = -0.520007849 Simplifying r + -0.642857143 = -0.520007849 Reorder the terms: -0.642857143 + r = -0.520007849 Solving -0.642857143 + r = -0.520007849 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.642857143' to each side of the equation. -0.642857143 + 0.642857143 + r = -0.520007849 + 0.642857143 Combine like terms: -0.642857143 + 0.642857143 = 0.000000000 0.000000000 + r = -0.520007849 + 0.642857143 r = -0.520007849 + 0.642857143 Combine like terms: -0.520007849 + 0.642857143 = 0.122849294 r = 0.122849294 Simplifying r = 0.122849294

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.162864992, 0.122849294}

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