2r^2+11r-15=0

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Solution for 2r^2+11r-15=0 equation:


Simplifying
2r2 + 11r + -15 = 0

Reorder the terms:
-15 + 11r + 2r2 = 0

Solving
-15 + 11r + 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'.
-7.5 + 5.5r + r2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + 5.5r + 7.5 + r2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + 5.5r + r2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + 5.5r + r2 = 0 + 7.5
5.5r + r2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
5.5r + r2 = 7.5

The r term is 5.5r.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5r + 7.5625 + r2 = 7.5 + 7.5625

Reorder the terms:
7.5625 + 5.5r + r2 = 7.5 + 7.5625

Combine like terms: 7.5 + 7.5625 = 15.0625
7.5625 + 5.5r + r2 = 15.0625

Factor a perfect square on the left side:
(r + 2.75)(r + 2.75) = 15.0625

Calculate the square root of the right side: 3.881043674

Break this problem into two subproblems by setting 
(r + 2.75) equal to 3.881043674 and -3.881043674.

Subproblem 1

r + 2.75 = 3.881043674 Simplifying r + 2.75 = 3.881043674 Reorder the terms: 2.75 + r = 3.881043674 Solving 2.75 + r = 3.881043674 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + r = 3.881043674 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + r = 3.881043674 + -2.75 r = 3.881043674 + -2.75 Combine like terms: 3.881043674 + -2.75 = 1.131043674 r = 1.131043674 Simplifying r = 1.131043674

Subproblem 2

r + 2.75 = -3.881043674 Simplifying r + 2.75 = -3.881043674 Reorder the terms: 2.75 + r = -3.881043674 Solving 2.75 + r = -3.881043674 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + r = -3.881043674 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + r = -3.881043674 + -2.75 r = -3.881043674 + -2.75 Combine like terms: -3.881043674 + -2.75 = -6.631043674 r = -6.631043674 Simplifying r = -6.631043674

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.131043674, -6.631043674}

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