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Simplifying (x + 6)(x + -5) = 76 Reorder the terms: (6 + x)(x + -5) = 76 Reorder the terms: (6 + x)(-5 + x) = 76 Multiply (6 + x) * (-5 + x) (6(-5 + x) + x(-5 + x)) = 76 ((-5 * 6 + x * 6) + x(-5 + x)) = 76 ((-30 + 6x) + x(-5 + x)) = 76 (-30 + 6x + (-5 * x + x * x)) = 76 (-30 + 6x + (-5x + x2)) = 76 Combine like terms: 6x + -5x = 1x (-30 + 1x + x2) = 76 Solving -30 + 1x + x2 = 76 Solving for variable 'x'. Reorder the terms: -30 + -76 + 1x + x2 = 76 + -76 Combine like terms: -30 + -76 = -106 -106 + 1x + x2 = 76 + -76 Combine like terms: 76 + -76 = 0 -106 + 1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '106' to each side of the equation. -106 + 1x + 106 + x2 = 0 + 106 Reorder the terms: -106 + 106 + 1x + x2 = 0 + 106 Combine like terms: -106 + 106 = 0 0 + 1x + x2 = 0 + 106 1x + x2 = 0 + 106 Combine like terms: 0 + 106 = 106 1x + x2 = 106 The x term is 1x. 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. x + 0.25 + x2 = 106 + 0.25 Reorder the terms: 0.25 + x + x2 = 106 + 0.25 Combine like terms: 106 + 0.25 = 106.25 0.25 + x + x2 = 106.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 106.25 Calculate the square root of the right side: 10.307764064 Break this problem into two subproblems by setting (x + 0.5) equal to 10.307764064 and -10.307764064.Subproblem 1
x + 0.5 = 10.307764064 Simplifying x + 0.5 = 10.307764064 Reorder the terms: 0.5 + x = 10.307764064 Solving 0.5 + x = 10.307764064 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 10.307764064 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 10.307764064 + -0.5 x = 10.307764064 + -0.5 Combine like terms: 10.307764064 + -0.5 = 9.807764064 x = 9.807764064 Simplifying x = 9.807764064Subproblem 2
x + 0.5 = -10.307764064 Simplifying x + 0.5 = -10.307764064 Reorder the terms: 0.5 + x = -10.307764064 Solving 0.5 + x = -10.307764064 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -10.307764064 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -10.307764064 + -0.5 x = -10.307764064 + -0.5 Combine like terms: -10.307764064 + -0.5 = -10.807764064 x = -10.807764064 Simplifying x = -10.807764064Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.807764064, -10.807764064}
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