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Simplifying 6r2 + -14r + -15 = 0 Reorder the terms: -15 + -14r + 6r2 = 0 Solving -15 + -14r + 6r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.5 + -2.333333333r + r2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -2.333333333r + 2.5 + r2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -2.333333333r + r2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -2.333333333r + r2 = 0 + 2.5 -2.333333333r + r2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -2.333333333r + r2 = 2.5 The r term is -2.333333333r. Take half its coefficient (-1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. -2.333333333r + 1.361111112 + r2 = 2.5 + 1.361111112 Reorder the terms: 1.361111112 + -2.333333333r + r2 = 2.5 + 1.361111112 Combine like terms: 2.5 + 1.361111112 = 3.861111112 1.361111112 + -2.333333333r + r2 = 3.861111112 Factor a perfect square on the left side: (r + -1.166666667)(r + -1.166666667) = 3.861111112 Calculate the square root of the right side: 1.964971021 Break this problem into two subproblems by setting (r + -1.166666667) equal to 1.964971021 and -1.964971021.Subproblem 1
r + -1.166666667 = 1.964971021 Simplifying r + -1.166666667 = 1.964971021 Reorder the terms: -1.166666667 + r = 1.964971021 Solving -1.166666667 + r = 1.964971021 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + r = 1.964971021 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + r = 1.964971021 + 1.166666667 r = 1.964971021 + 1.166666667 Combine like terms: 1.964971021 + 1.166666667 = 3.131637688 r = 3.131637688 Simplifying r = 3.131637688Subproblem 2
r + -1.166666667 = -1.964971021 Simplifying r + -1.166666667 = -1.964971021 Reorder the terms: -1.166666667 + r = -1.964971021 Solving -1.166666667 + r = -1.964971021 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + r = -1.964971021 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + r = -1.964971021 + 1.166666667 r = -1.964971021 + 1.166666667 Combine like terms: -1.964971021 + 1.166666667 = -0.798304354 r = -0.798304354 Simplifying r = -0.798304354Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.131637688, -0.798304354}
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