2r^2+15r-19=0

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Solution for 2r^2+15r-19=0 equation:


Simplifying
2r2 + 15r + -19 = 0

Reorder the terms:
-19 + 15r + 2r2 = 0

Solving
-19 + 15r + 2r2 = 0

Solving for variable 'r'.

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

Divide each side by '2'.
-9.5 + 7.5r + r2 = 0

Move the constant term to the right:

Add '9.5' to each side of the equation.
-9.5 + 7.5r + 9.5 + r2 = 0 + 9.5

Reorder the terms:
-9.5 + 9.5 + 7.5r + r2 = 0 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + 7.5r + r2 = 0 + 9.5
7.5r + r2 = 0 + 9.5

Combine like terms: 0 + 9.5 = 9.5
7.5r + r2 = 9.5

The r term is 7.5r.  Take half its coefficient (3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
7.5r + 14.0625 + r2 = 9.5 + 14.0625

Reorder the terms:
14.0625 + 7.5r + r2 = 9.5 + 14.0625

Combine like terms: 9.5 + 14.0625 = 23.5625
14.0625 + 7.5r + r2 = 23.5625

Factor a perfect square on the left side:
(r + 3.75)(r + 3.75) = 23.5625

Calculate the square root of the right side: 4.85412196

Break this problem into two subproblems by setting 
(r + 3.75) equal to 4.85412196 and -4.85412196.

Subproblem 1

r + 3.75 = 4.85412196 Simplifying r + 3.75 = 4.85412196 Reorder the terms: 3.75 + r = 4.85412196 Solving 3.75 + r = 4.85412196 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + r = 4.85412196 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + r = 4.85412196 + -3.75 r = 4.85412196 + -3.75 Combine like terms: 4.85412196 + -3.75 = 1.10412196 r = 1.10412196 Simplifying r = 1.10412196

Subproblem 2

r + 3.75 = -4.85412196 Simplifying r + 3.75 = -4.85412196 Reorder the terms: 3.75 + r = -4.85412196 Solving 3.75 + r = -4.85412196 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + r = -4.85412196 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + r = -4.85412196 + -3.75 r = -4.85412196 + -3.75 Combine like terms: -4.85412196 + -3.75 = -8.60412196 r = -8.60412196 Simplifying r = -8.60412196

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.10412196, -8.60412196}

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