5r^2+21r-10=0

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Solution for 5r^2+21r-10=0 equation:


Simplifying
5r2 + 21r + -10 = 0

Reorder the terms:
-10 + 21r + 5r2 = 0

Solving
-10 + 21r + 5r2 = 0

Solving for variable 'r'.

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

Divide each side by '5'.
-2 + 4.2r + r2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 4.2r + 2 + r2 = 0 + 2

Reorder the terms:
-2 + 2 + 4.2r + r2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 4.2r + r2 = 0 + 2
4.2r + r2 = 0 + 2

Combine like terms: 0 + 2 = 2
4.2r + r2 = 2

The r term is 4.2r.  Take half its coefficient (2.1).
Square it (4.41) and add it to both sides.

Add '4.41' to each side of the equation.
4.2r + 4.41 + r2 = 2 + 4.41

Reorder the terms:
4.41 + 4.2r + r2 = 2 + 4.41

Combine like terms: 2 + 4.41 = 6.41
4.41 + 4.2r + r2 = 6.41

Factor a perfect square on the left side:
(r + 2.1)(r + 2.1) = 6.41

Calculate the square root of the right side: 2.53179778

Break this problem into two subproblems by setting 
(r + 2.1) equal to 2.53179778 and -2.53179778.

Subproblem 1

r + 2.1 = 2.53179778 Simplifying r + 2.1 = 2.53179778 Reorder the terms: 2.1 + r = 2.53179778 Solving 2.1 + r = 2.53179778 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.1' to each side of the equation. 2.1 + -2.1 + r = 2.53179778 + -2.1 Combine like terms: 2.1 + -2.1 = 0.0 0.0 + r = 2.53179778 + -2.1 r = 2.53179778 + -2.1 Combine like terms: 2.53179778 + -2.1 = 0.43179778 r = 0.43179778 Simplifying r = 0.43179778

Subproblem 2

r + 2.1 = -2.53179778 Simplifying r + 2.1 = -2.53179778 Reorder the terms: 2.1 + r = -2.53179778 Solving 2.1 + r = -2.53179778 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.1' to each side of the equation. 2.1 + -2.1 + r = -2.53179778 + -2.1 Combine like terms: 2.1 + -2.1 = 0.0 0.0 + r = -2.53179778 + -2.1 r = -2.53179778 + -2.1 Combine like terms: -2.53179778 + -2.1 = -4.63179778 r = -4.63179778 Simplifying r = -4.63179778

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.43179778, -4.63179778}

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