14093.89=628r^2+617.952r

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Solution for 14093.89=628r^2+617.952r equation:


Simplifying
14093.89 = 628r2 + 617.952r

Reorder the terms:
14093.89 = 617.952r + 628r2

Solving
14093.89 = 617.952r + 628r2

Solving for variable 'r'.

Reorder the terms:
14093.89 + -617.952r + -628r2 = 617.952r + -617.952r + 628r2 + -628r2

Combine like terms: 617.952r + -617.952r = 0.000
14093.89 + -617.952r + -628r2 = 0.000 + 628r2 + -628r2
14093.89 + -617.952r + -628r2 = 628r2 + -628r2

Combine like terms: 628r2 + -628r2 = 0
14093.89 + -617.952r + -628r2 = 0

Begin completing the square.  Divide all terms by
-628 the coefficient of the squared term: 

Divide each side by '-628'.
-22.4425 + 0.984r + r2 = 0

Move the constant term to the right:

Add '22.4425' to each side of the equation.
-22.4425 + 0.984r + 22.4425 + r2 = 0 + 22.4425

Reorder the terms:
-22.4425 + 22.4425 + 0.984r + r2 = 0 + 22.4425

Combine like terms: -22.4425 + 22.4425 = 0.0000
0.0000 + 0.984r + r2 = 0 + 22.4425
0.984r + r2 = 0 + 22.4425

Combine like terms: 0 + 22.4425 = 22.4425
0.984r + r2 = 22.4425

The r term is 0.984r.  Take half its coefficient (0.492).
Square it (0.242064) and add it to both sides.

Add '0.242064' to each side of the equation.
0.984r + 0.242064 + r2 = 22.4425 + 0.242064

Reorder the terms:
0.242064 + 0.984r + r2 = 22.4425 + 0.242064

Combine like terms: 22.4425 + 0.242064 = 22.684564
0.242064 + 0.984r + r2 = 22.684564

Factor a perfect square on the left side:
(r + 0.492)(r + 0.492) = 22.684564

Calculate the square root of the right side: 4.762831511

Break this problem into two subproblems by setting 
(r + 0.492) equal to 4.762831511 and -4.762831511.

Subproblem 1

r + 0.492 = 4.762831511 Simplifying r + 0.492 = 4.762831511 Reorder the terms: 0.492 + r = 4.762831511 Solving 0.492 + r = 4.762831511 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.492' to each side of the equation. 0.492 + -0.492 + r = 4.762831511 + -0.492 Combine like terms: 0.492 + -0.492 = 0.000 0.000 + r = 4.762831511 + -0.492 r = 4.762831511 + -0.492 Combine like terms: 4.762831511 + -0.492 = 4.270831511 r = 4.270831511 Simplifying r = 4.270831511

Subproblem 2

r + 0.492 = -4.762831511 Simplifying r + 0.492 = -4.762831511 Reorder the terms: 0.492 + r = -4.762831511 Solving 0.492 + r = -4.762831511 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.492' to each side of the equation. 0.492 + -0.492 + r = -4.762831511 + -0.492 Combine like terms: 0.492 + -0.492 = 0.000 0.000 + r = -4.762831511 + -0.492 r = -4.762831511 + -0.492 Combine like terms: -4.762831511 + -0.492 = -5.254831511 r = -5.254831511 Simplifying r = -5.254831511

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.270831511, -5.254831511}

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