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Simplifying x(x + 5) = 106 Reorder the terms: x(5 + x) = 106 (5 * x + x * x) = 106 (5x + x2) = 106 Solving 5x + x2 = 106 Solving for variable 'x'. Reorder the terms: -106 + 5x + x2 = 106 + -106 Combine like terms: 106 + -106 = 0 -106 + 5x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '106' to each side of the equation. -106 + 5x + 106 + x2 = 0 + 106 Reorder the terms: -106 + 106 + 5x + x2 = 0 + 106 Combine like terms: -106 + 106 = 0 0 + 5x + x2 = 0 + 106 5x + x2 = 0 + 106 Combine like terms: 0 + 106 = 106 5x + x2 = 106 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = 106 + 6.25 Reorder the terms: 6.25 + 5x + x2 = 106 + 6.25 Combine like terms: 106 + 6.25 = 112.25 6.25 + 5x + x2 = 112.25 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 112.25 Calculate the square root of the right side: 10.59481005 Break this problem into two subproblems by setting (x + 2.5) equal to 10.59481005 and -10.59481005.Subproblem 1
x + 2.5 = 10.59481005 Simplifying x + 2.5 = 10.59481005 Reorder the terms: 2.5 + x = 10.59481005 Solving 2.5 + x = 10.59481005 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 10.59481005 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 10.59481005 + -2.5 x = 10.59481005 + -2.5 Combine like terms: 10.59481005 + -2.5 = 8.09481005 x = 8.09481005 Simplifying x = 8.09481005Subproblem 2
x + 2.5 = -10.59481005 Simplifying x + 2.5 = -10.59481005 Reorder the terms: 2.5 + x = -10.59481005 Solving 2.5 + x = -10.59481005 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -10.59481005 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -10.59481005 + -2.5 x = -10.59481005 + -2.5 Combine like terms: -10.59481005 + -2.5 = -13.09481005 x = -13.09481005 Simplifying x = -13.09481005Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.09481005, -13.09481005}
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