6r^2=28r+16

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Solution for 6r^2=28r+16 equation:


Simplifying
6r2 = 28r + 16

Reorder the terms:
6r2 = 16 + 28r

Solving
6r2 = 16 + 28r

Solving for variable 'r'.

Reorder the terms:
-16 + -28r + 6r2 = 16 + 28r + -16 + -28r

Reorder the terms:
-16 + -28r + 6r2 = 16 + -16 + 28r + -28r

Combine like terms: 16 + -16 = 0
-16 + -28r + 6r2 = 0 + 28r + -28r
-16 + -28r + 6r2 = 28r + -28r

Combine like terms: 28r + -28r = 0
-16 + -28r + 6r2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-8 + -14r + 3r2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(-8 + -14r + 3r2)' equal to zero and attempt to solve: Simplifying -8 + -14r + 3r2 = 0 Solving -8 + -14r + 3r2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2.666666667 + -4.666666667r + r2 = 0 Move the constant term to the right: Add '2.666666667' to each side of the equation. -2.666666667 + -4.666666667r + 2.666666667 + r2 = 0 + 2.666666667 Reorder the terms: -2.666666667 + 2.666666667 + -4.666666667r + r2 = 0 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + -4.666666667r + r2 = 0 + 2.666666667 -4.666666667r + r2 = 0 + 2.666666667 Combine like terms: 0 + 2.666666667 = 2.666666667 -4.666666667r + r2 = 2.666666667 The r term is -4.666666667r. Take half its coefficient (-2.333333334). Square it (5.444444448) and add it to both sides. Add '5.444444448' to each side of the equation. -4.666666667r + 5.444444448 + r2 = 2.666666667 + 5.444444448 Reorder the terms: 5.444444448 + -4.666666667r + r2 = 2.666666667 + 5.444444448 Combine like terms: 2.666666667 + 5.444444448 = 8.111111115 5.444444448 + -4.666666667r + r2 = 8.111111115 Factor a perfect square on the left side: (r + -2.333333334)(r + -2.333333334) = 8.111111115 Calculate the square root of the right side: 2.848001249 Break this problem into two subproblems by setting (r + -2.333333334) equal to 2.848001249 and -2.848001249.

Subproblem 1

r + -2.333333334 = 2.848001249 Simplifying r + -2.333333334 = 2.848001249 Reorder the terms: -2.333333334 + r = 2.848001249 Solving -2.333333334 + r = 2.848001249 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + r = 2.848001249 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + r = 2.848001249 + 2.333333334 r = 2.848001249 + 2.333333334 Combine like terms: 2.848001249 + 2.333333334 = 5.181334583 r = 5.181334583 Simplifying r = 5.181334583

Subproblem 2

r + -2.333333334 = -2.848001249 Simplifying r + -2.333333334 = -2.848001249 Reorder the terms: -2.333333334 + r = -2.848001249 Solving -2.333333334 + r = -2.848001249 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + r = -2.848001249 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + r = -2.848001249 + 2.333333334 r = -2.848001249 + 2.333333334 Combine like terms: -2.848001249 + 2.333333334 = -0.514667915 r = -0.514667915 Simplifying r = -0.514667915

Solution

The solution to the problem is based on the solutions from the subproblems. r = {5.181334583, -0.514667915}

Solution

r = {5.181334583, -0.514667915}

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