2r^2=7r-4

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Solution for 2r^2=7r-4 equation:


Simplifying
2r2 = 7r + -4

Reorder the terms:
2r2 = -4 + 7r

Solving
2r2 = -4 + 7r

Solving for variable 'r'.

Reorder the terms:
4 + -7r + 2r2 = -4 + 7r + 4 + -7r

Reorder the terms:
4 + -7r + 2r2 = -4 + 4 + 7r + -7r

Combine like terms: -4 + 4 = 0
4 + -7r + 2r2 = 0 + 7r + -7r
4 + -7r + 2r2 = 7r + -7r

Combine like terms: 7r + -7r = 0
4 + -7r + 2r2 = 0

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

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

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -3.5r + -2 + r2 = 0 + -2

Reorder the terms:
2 + -2 + -3.5r + r2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -3.5r + r2 = 0 + -2
-3.5r + r2 = 0 + -2

Combine like terms: 0 + -2 = -2
-3.5r + r2 = -2

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 = -2 + 3.0625

Reorder the terms:
3.0625 + -3.5r + r2 = -2 + 3.0625

Combine like terms: -2 + 3.0625 = 1.0625
3.0625 + -3.5r + r2 = 1.0625

Factor a perfect square on the left side:
(r + -1.75)(r + -1.75) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(r + -1.75) equal to 1.030776406 and -1.030776406.

Subproblem 1

r + -1.75 = 1.030776406 Simplifying r + -1.75 = 1.030776406 Reorder the terms: -1.75 + r = 1.030776406 Solving -1.75 + r = 1.030776406 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 = 1.030776406 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + r = 1.030776406 + 1.75 r = 1.030776406 + 1.75 Combine like terms: 1.030776406 + 1.75 = 2.780776406 r = 2.780776406 Simplifying r = 2.780776406

Subproblem 2

r + -1.75 = -1.030776406 Simplifying r + -1.75 = -1.030776406 Reorder the terms: -1.75 + r = -1.030776406 Solving -1.75 + r = -1.030776406 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 = -1.030776406 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + r = -1.030776406 + 1.75 r = -1.030776406 + 1.75 Combine like terms: -1.030776406 + 1.75 = 0.719223594 r = 0.719223594 Simplifying r = 0.719223594

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.780776406, 0.719223594}

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