2r^2-7r=1

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Solution for 2r^2-7r=1 equation:


Simplifying
2r2 + -7r = 1

Reorder the terms:
-7r + 2r2 = 1

Solving
-7r + 2r2 = 1

Solving for variable 'r'.

Reorder the terms:
-1 + -7r + 2r2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -7r + 2r2 = 0

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

Divide each side by '2'.
-0.5 + -3.5r + r2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -3.5r + 0.5 + r2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -3.5r + r2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -3.5r + r2 = 0 + 0.5
-3.5r + r2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-3.5r + r2 = 0.5

The r term is -3.5r.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5r + 3.0625 + r2 = 0.5 + 3.0625

Reorder the terms:
3.0625 + -3.5r + r2 = 0.5 + 3.0625

Combine like terms: 0.5 + 3.0625 = 3.5625
3.0625 + -3.5r + r2 = 3.5625

Factor a perfect square on the left side:
(r + -1.75)(r + -1.75) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(r + -1.75) equal to 1.887458609 and -1.887458609.

Subproblem 1

r + -1.75 = 1.887458609 Simplifying r + -1.75 = 1.887458609 Reorder the terms: -1.75 + r = 1.887458609 Solving -1.75 + r = 1.887458609 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + r = 1.887458609 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + r = 1.887458609 + 1.75 r = 1.887458609 + 1.75 Combine like terms: 1.887458609 + 1.75 = 3.637458609 r = 3.637458609 Simplifying r = 3.637458609

Subproblem 2

r + -1.75 = -1.887458609 Simplifying r + -1.75 = -1.887458609 Reorder the terms: -1.75 + r = -1.887458609 Solving -1.75 + r = -1.887458609 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + r = -1.887458609 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + r = -1.887458609 + 1.75 r = -1.887458609 + 1.75 Combine like terms: -1.887458609 + 1.75 = -0.137458609 r = -0.137458609 Simplifying r = -0.137458609

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.637458609, -0.137458609}

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