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Simplifying 4r2 + 16r + -8 + 7r2 + -4r + 4 + r2 + -8r + -5 = 0 Reorder the terms: -8 + 4 + -5 + 16r + -4r + -8r + 4r2 + 7r2 + r2 = 0 Combine like terms: -8 + 4 = -4 -4 + -5 + 16r + -4r + -8r + 4r2 + 7r2 + r2 = 0 Combine like terms: -4 + -5 = -9 -9 + 16r + -4r + -8r + 4r2 + 7r2 + r2 = 0 Combine like terms: 16r + -4r = 12r -9 + 12r + -8r + 4r2 + 7r2 + r2 = 0 Combine like terms: 12r + -8r = 4r -9 + 4r + 4r2 + 7r2 + r2 = 0 Combine like terms: 4r2 + 7r2 = 11r2 -9 + 4r + 11r2 + r2 = 0 Combine like terms: 11r2 + r2 = 12r2 -9 + 4r + 12r2 = 0 Solving -9 + 4r + 12r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.75 + 0.3333333333r + r2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.3333333333r + 0.75 + r2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.3333333333r + r2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.3333333333r + r2 = 0 + 0.75 0.3333333333r + r2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.3333333333r + r2 = 0.75 The r term is 0.3333333333r. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333r + 0.02777777779 + r2 = 0.75 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333r + r2 = 0.75 + 0.02777777779 Combine like terms: 0.75 + 0.02777777779 = 0.77777777779 0.02777777779 + 0.3333333333r + r2 = 0.77777777779 Factor a perfect square on the left side: (r + 0.1666666667)(r + 0.1666666667) = 0.77777777779 Calculate the square root of the right side: 0.881917104 Break this problem into two subproblems by setting (r + 0.1666666667) equal to 0.881917104 and -0.881917104.Subproblem 1
r + 0.1666666667 = 0.881917104 Simplifying r + 0.1666666667 = 0.881917104 Reorder the terms: 0.1666666667 + r = 0.881917104 Solving 0.1666666667 + r = 0.881917104 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + r = 0.881917104 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + r = 0.881917104 + -0.1666666667 r = 0.881917104 + -0.1666666667 Combine like terms: 0.881917104 + -0.1666666667 = 0.7152504373 r = 0.7152504373 Simplifying r = 0.7152504373Subproblem 2
r + 0.1666666667 = -0.881917104 Simplifying r + 0.1666666667 = -0.881917104 Reorder the terms: 0.1666666667 + r = -0.881917104 Solving 0.1666666667 + r = -0.881917104 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + r = -0.881917104 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + r = -0.881917104 + -0.1666666667 r = -0.881917104 + -0.1666666667 Combine like terms: -0.881917104 + -0.1666666667 = -1.0485837707 r = -1.0485837707 Simplifying r = -1.0485837707Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.7152504373, -1.0485837707}
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