15r^2-34r+-15=0

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Solution for 15r^2-34r+-15=0 equation:


Simplifying
15r2 + -34r + -15 = 0

Reorder the terms:
-15 + -34r + 15r2 = 0

Solving
-15 + -34r + 15r2 = 0

Solving for variable 'r'.

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

Divide each side by '15'.
-1 + -2.266666667r + r2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2.266666667r + 1 + r2 = 0 + 1

Reorder the terms:
-1 + 1 + -2.266666667r + r2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2.266666667r + r2 = 0 + 1
-2.266666667r + r2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2.266666667r + r2 = 1

The r term is -2.266666667r.  Take half its coefficient (-1.133333334).
Square it (1.284444446) and add it to both sides.

Add '1.284444446' to each side of the equation.
-2.266666667r + 1.284444446 + r2 = 1 + 1.284444446

Reorder the terms:
1.284444446 + -2.266666667r + r2 = 1 + 1.284444446

Combine like terms: 1 + 1.284444446 = 2.284444446
1.284444446 + -2.266666667r + r2 = 2.284444446

Factor a perfect square on the left side:
(r + -1.133333334)(r + -1.133333334) = 2.284444446

Calculate the square root of the right side: 1.511437874

Break this problem into two subproblems by setting 
(r + -1.133333334) equal to 1.511437874 and -1.511437874.

Subproblem 1

r + -1.133333334 = 1.511437874 Simplifying r + -1.133333334 = 1.511437874 Reorder the terms: -1.133333334 + r = 1.511437874 Solving -1.133333334 + r = 1.511437874 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.133333334' to each side of the equation. -1.133333334 + 1.133333334 + r = 1.511437874 + 1.133333334 Combine like terms: -1.133333334 + 1.133333334 = 0.000000000 0.000000000 + r = 1.511437874 + 1.133333334 r = 1.511437874 + 1.133333334 Combine like terms: 1.511437874 + 1.133333334 = 2.644771208 r = 2.644771208 Simplifying r = 2.644771208

Subproblem 2

r + -1.133333334 = -1.511437874 Simplifying r + -1.133333334 = -1.511437874 Reorder the terms: -1.133333334 + r = -1.511437874 Solving -1.133333334 + r = -1.511437874 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.133333334' to each side of the equation. -1.133333334 + 1.133333334 + r = -1.511437874 + 1.133333334 Combine like terms: -1.133333334 + 1.133333334 = 0.000000000 0.000000000 + r = -1.511437874 + 1.133333334 r = -1.511437874 + 1.133333334 Combine like terms: -1.511437874 + 1.133333334 = -0.37810454 r = -0.37810454 Simplifying r = -0.37810454

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.644771208, -0.37810454}

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