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Simplifying r2 + -1r + 2 = 7 Reorder the terms: 2 + -1r + r2 = 7 Solving 2 + -1r + r2 = 7 Solving for variable 'r'. Reorder the terms: 2 + -7 + -1r + r2 = 7 + -7 Combine like terms: 2 + -7 = -5 -5 + -1r + r2 = 7 + -7 Combine like terms: 7 + -7 = 0 -5 + -1r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + -1r + 5 + r2 = 0 + 5 Reorder the terms: -5 + 5 + -1r + r2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -1r + r2 = 0 + 5 -1r + r2 = 0 + 5 Combine like terms: 0 + 5 = 5 -1r + r2 = 5 The r term is -1r. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1r + 0.25 + r2 = 5 + 0.25 Reorder the terms: 0.25 + -1r + r2 = 5 + 0.25 Combine like terms: 5 + 0.25 = 5.25 0.25 + -1r + r2 = 5.25 Factor a perfect square on the left side: (r + -0.5)(r + -0.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (r + -0.5) equal to 2.291287847 and -2.291287847.Subproblem 1
r + -0.5 = 2.291287847 Simplifying r + -0.5 = 2.291287847 Reorder the terms: -0.5 + r = 2.291287847 Solving -0.5 + r = 2.291287847 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + r = 2.291287847 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = 2.291287847 + 0.5 r = 2.291287847 + 0.5 Combine like terms: 2.291287847 + 0.5 = 2.791287847 r = 2.791287847 Simplifying r = 2.791287847Subproblem 2
r + -0.5 = -2.291287847 Simplifying r + -0.5 = -2.291287847 Reorder the terms: -0.5 + r = -2.291287847 Solving -0.5 + r = -2.291287847 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + r = -2.291287847 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = -2.291287847 + 0.5 r = -2.291287847 + 0.5 Combine like terms: -2.291287847 + 0.5 = -1.791287847 r = -1.791287847 Simplifying r = -1.791287847Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.791287847, -1.791287847}
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