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Simplifying (x + 5)(x + 7) = 100 Reorder the terms: (5 + x)(x + 7) = 100 Reorder the terms: (5 + x)(7 + x) = 100 Multiply (5 + x) * (7 + x) (5(7 + x) + x(7 + x)) = 100 ((7 * 5 + x * 5) + x(7 + x)) = 100 ((35 + 5x) + x(7 + x)) = 100 (35 + 5x + (7 * x + x * x)) = 100 (35 + 5x + (7x + x2)) = 100 Combine like terms: 5x + 7x = 12x (35 + 12x + x2) = 100 Solving 35 + 12x + x2 = 100 Solving for variable 'x'. Reorder the terms: 35 + -100 + 12x + x2 = 100 + -100 Combine like terms: 35 + -100 = -65 -65 + 12x + x2 = 100 + -100 Combine like terms: 100 + -100 = 0 -65 + 12x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '65' to each side of the equation. -65 + 12x + 65 + x2 = 0 + 65 Reorder the terms: -65 + 65 + 12x + x2 = 0 + 65 Combine like terms: -65 + 65 = 0 0 + 12x + x2 = 0 + 65 12x + x2 = 0 + 65 Combine like terms: 0 + 65 = 65 12x + x2 = 65 The x term is 12x. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12x + 36 + x2 = 65 + 36 Reorder the terms: 36 + 12x + x2 = 65 + 36 Combine like terms: 65 + 36 = 101 36 + 12x + x2 = 101 Factor a perfect square on the left side: (x + 6)(x + 6) = 101 Calculate the square root of the right side: 10.049875621 Break this problem into two subproblems by setting (x + 6) equal to 10.049875621 and -10.049875621.Subproblem 1
x + 6 = 10.049875621 Simplifying x + 6 = 10.049875621 Reorder the terms: 6 + x = 10.049875621 Solving 6 + x = 10.049875621 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + x = 10.049875621 + -6 Combine like terms: 6 + -6 = 0 0 + x = 10.049875621 + -6 x = 10.049875621 + -6 Combine like terms: 10.049875621 + -6 = 4.049875621 x = 4.049875621 Simplifying x = 4.049875621Subproblem 2
x + 6 = -10.049875621 Simplifying x + 6 = -10.049875621 Reorder the terms: 6 + x = -10.049875621 Solving 6 + x = -10.049875621 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + x = -10.049875621 + -6 Combine like terms: 6 + -6 = 0 0 + x = -10.049875621 + -6 x = -10.049875621 + -6 Combine like terms: -10.049875621 + -6 = -16.049875621 x = -16.049875621 Simplifying x = -16.049875621Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.049875621, -16.049875621}
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