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Simplifying 6r2 + -16r + -54 = 0 Reorder the terms: -54 + -16r + 6r2 = 0 Solving -54 + -16r + 6r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(-27 + -8r + 3r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-27 + -8r + 3r2)' equal to zero and attempt to solve: Simplifying -27 + -8r + 3r2 = 0 Solving -27 + -8r + 3r2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -9 + -2.666666667r + r2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + -2.666666667r + 9 + r2 = 0 + 9 Reorder the terms: -9 + 9 + -2.666666667r + r2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -2.666666667r + r2 = 0 + 9 -2.666666667r + r2 = 0 + 9 Combine like terms: 0 + 9 = 9 -2.666666667r + r2 = 9 The r term is -2.666666667r. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667r + 1.777777780 + r2 = 9 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667r + r2 = 9 + 1.777777780 Combine like terms: 9 + 1.777777780 = 10.77777778 1.777777780 + -2.666666667r + r2 = 10.77777778 Factor a perfect square on the left side: (r + -1.333333334)(r + -1.333333334) = 10.77777778 Calculate the square root of the right side: 3.282952601 Break this problem into two subproblems by setting (r + -1.333333334) equal to 3.282952601 and -3.282952601.Subproblem 1
r + -1.333333334 = 3.282952601 Simplifying r + -1.333333334 = 3.282952601 Reorder the terms: -1.333333334 + r = 3.282952601 Solving -1.333333334 + r = 3.282952601 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + r = 3.282952601 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + r = 3.282952601 + 1.333333334 r = 3.282952601 + 1.333333334 Combine like terms: 3.282952601 + 1.333333334 = 4.616285935 r = 4.616285935 Simplifying r = 4.616285935Subproblem 2
r + -1.333333334 = -3.282952601 Simplifying r + -1.333333334 = -3.282952601 Reorder the terms: -1.333333334 + r = -3.282952601 Solving -1.333333334 + r = -3.282952601 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + r = -3.282952601 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + r = -3.282952601 + 1.333333334 r = -3.282952601 + 1.333333334 Combine like terms: -3.282952601 + 1.333333334 = -1.949619267 r = -1.949619267 Simplifying r = -1.949619267Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.616285935, -1.949619267}Solution
r = {4.616285935, -1.949619267}
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