6r^2+46r-16=11

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Solution for 6r^2+46r-16=11 equation:


Simplifying
6r2 + 46r + -16 = 11

Reorder the terms:
-16 + 46r + 6r2 = 11

Solving
-16 + 46r + 6r2 = 11

Solving for variable 'r'.

Reorder the terms:
-16 + -11 + 46r + 6r2 = 11 + -11

Combine like terms: -16 + -11 = -27
-27 + 46r + 6r2 = 11 + -11

Combine like terms: 11 + -11 = 0
-27 + 46r + 6r2 = 0

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

Divide each side by '6'.
-4.5 + 7.666666667r + r2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 7.666666667r + 4.5 + r2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 7.666666667r + r2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 7.666666667r + r2 = 0 + 4.5
7.666666667r + r2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
7.666666667r + r2 = 4.5

The r term is 7.666666667r.  Take half its coefficient (3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
7.666666667r + 14.69444445 + r2 = 4.5 + 14.69444445

Reorder the terms:
14.69444445 + 7.666666667r + r2 = 4.5 + 14.69444445

Combine like terms: 4.5 + 14.69444445 = 19.19444445
14.69444445 + 7.666666667r + r2 = 19.19444445

Factor a perfect square on the left side:
(r + 3.833333334)(r + 3.833333334) = 19.19444445

Calculate the square root of the right side: 4.381146477

Break this problem into two subproblems by setting 
(r + 3.833333334) equal to 4.381146477 and -4.381146477.

Subproblem 1

r + 3.833333334 = 4.381146477 Simplifying r + 3.833333334 = 4.381146477 Reorder the terms: 3.833333334 + r = 4.381146477 Solving 3.833333334 + r = 4.381146477 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + r = 4.381146477 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + r = 4.381146477 + -3.833333334 r = 4.381146477 + -3.833333334 Combine like terms: 4.381146477 + -3.833333334 = 0.547813143 r = 0.547813143 Simplifying r = 0.547813143

Subproblem 2

r + 3.833333334 = -4.381146477 Simplifying r + 3.833333334 = -4.381146477 Reorder the terms: 3.833333334 + r = -4.381146477 Solving 3.833333334 + r = -4.381146477 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + r = -4.381146477 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + r = -4.381146477 + -3.833333334 r = -4.381146477 + -3.833333334 Combine like terms: -4.381146477 + -3.833333334 = -8.214479811 r = -8.214479811 Simplifying r = -8.214479811

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.547813143, -8.214479811}

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