25r^2+4=-21r

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Solution for 25r^2+4=-21r equation:


Simplifying
25r2 + 4 = -21r

Reorder the terms:
4 + 25r2 = -21r

Solving
4 + 25r2 = -21r

Solving for variable 'r'.

Reorder the terms:
4 + 21r + 25r2 = -21r + 21r

Combine like terms: -21r + 21r = 0
4 + 21r + 25r2 = 0

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

Divide each side by '25'.
0.16 + 0.84r + r2 = 0

Move the constant term to the right:

Add '-0.16' to each side of the equation.
0.16 + 0.84r + -0.16 + r2 = 0 + -0.16

Reorder the terms:
0.16 + -0.16 + 0.84r + r2 = 0 + -0.16

Combine like terms: 0.16 + -0.16 = 0.00
0.00 + 0.84r + r2 = 0 + -0.16
0.84r + r2 = 0 + -0.16

Combine like terms: 0 + -0.16 = -0.16
0.84r + r2 = -0.16

The r term is 0.84r.  Take half its coefficient (0.42).
Square it (0.1764) and add it to both sides.

Add '0.1764' to each side of the equation.
0.84r + 0.1764 + r2 = -0.16 + 0.1764

Reorder the terms:
0.1764 + 0.84r + r2 = -0.16 + 0.1764

Combine like terms: -0.16 + 0.1764 = 0.0164
0.1764 + 0.84r + r2 = 0.0164

Factor a perfect square on the left side:
(r + 0.42)(r + 0.42) = 0.0164

Calculate the square root of the right side: 0.128062485

Break this problem into two subproblems by setting 
(r + 0.42) equal to 0.128062485 and -0.128062485.

Subproblem 1

r + 0.42 = 0.128062485 Simplifying r + 0.42 = 0.128062485 Reorder the terms: 0.42 + r = 0.128062485 Solving 0.42 + r = 0.128062485 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.42' to each side of the equation. 0.42 + -0.42 + r = 0.128062485 + -0.42 Combine like terms: 0.42 + -0.42 = 0.00 0.00 + r = 0.128062485 + -0.42 r = 0.128062485 + -0.42 Combine like terms: 0.128062485 + -0.42 = -0.291937515 r = -0.291937515 Simplifying r = -0.291937515

Subproblem 2

r + 0.42 = -0.128062485 Simplifying r + 0.42 = -0.128062485 Reorder the terms: 0.42 + r = -0.128062485 Solving 0.42 + r = -0.128062485 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.42' to each side of the equation. 0.42 + -0.42 + r = -0.128062485 + -0.42 Combine like terms: 0.42 + -0.42 = 0.00 0.00 + r = -0.128062485 + -0.42 r = -0.128062485 + -0.42 Combine like terms: -0.128062485 + -0.42 = -0.548062485 r = -0.548062485 Simplifying r = -0.548062485

Solution

The solution to the problem is based on the solutions from the subproblems. r = {-0.291937515, -0.548062485}

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