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Simplifying 0 = 4r2 + -21r + 24 Reorder the terms: 0 = 24 + -21r + 4r2 Solving 0 = 24 + -21r + 4r2 Solving for variable 'r'. Combine like terms: 0 + -24 = -24 -24 + 21r + -4r2 = 24 + -21r + 4r2 + -24 + 21r + -4r2 Reorder the terms: -24 + 21r + -4r2 = 24 + -24 + -21r + 21r + 4r2 + -4r2 Combine like terms: 24 + -24 = 0 -24 + 21r + -4r2 = 0 + -21r + 21r + 4r2 + -4r2 -24 + 21r + -4r2 = -21r + 21r + 4r2 + -4r2 Combine like terms: -21r + 21r = 0 -24 + 21r + -4r2 = 0 + 4r2 + -4r2 -24 + 21r + -4r2 = 4r2 + -4r2 Combine like terms: 4r2 + -4r2 = 0 -24 + 21r + -4r2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 6 + -5.25r + r2 = 0 Move the constant term to the right: Add '-6' to each side of the equation. 6 + -5.25r + -6 + r2 = 0 + -6 Reorder the terms: 6 + -6 + -5.25r + r2 = 0 + -6 Combine like terms: 6 + -6 = 0 0 + -5.25r + r2 = 0 + -6 -5.25r + r2 = 0 + -6 Combine like terms: 0 + -6 = -6 -5.25r + r2 = -6 The r term is -5.25r. Take half its coefficient (-2.625). Square it (6.890625) and add it to both sides. Add '6.890625' to each side of the equation. -5.25r + 6.890625 + r2 = -6 + 6.890625 Reorder the terms: 6.890625 + -5.25r + r2 = -6 + 6.890625 Combine like terms: -6 + 6.890625 = 0.890625 6.890625 + -5.25r + r2 = 0.890625 Factor a perfect square on the left side: (r + -2.625)(r + -2.625) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (r + -2.625) equal to 0.943729304 and -0.943729304.Subproblem 1
r + -2.625 = 0.943729304 Simplifying r + -2.625 = 0.943729304 Reorder the terms: -2.625 + r = 0.943729304 Solving -2.625 + r = 0.943729304 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + r = 0.943729304 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + r = 0.943729304 + 2.625 r = 0.943729304 + 2.625 Combine like terms: 0.943729304 + 2.625 = 3.568729304 r = 3.568729304 Simplifying r = 3.568729304Subproblem 2
r + -2.625 = -0.943729304 Simplifying r + -2.625 = -0.943729304 Reorder the terms: -2.625 + r = -0.943729304 Solving -2.625 + r = -0.943729304 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + r = -0.943729304 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + r = -0.943729304 + 2.625 r = -0.943729304 + 2.625 Combine like terms: -0.943729304 + 2.625 = 1.681270696 r = 1.681270696 Simplifying r = 1.681270696Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.568729304, 1.681270696}
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