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Simplifying (x + 5)(x + 7) = 105 Reorder the terms: (5 + x)(x + 7) = 105 Reorder the terms: (5 + x)(7 + x) = 105 Multiply (5 + x) * (7 + x) (5(7 + x) + x(7 + x)) = 105 ((7 * 5 + x * 5) + x(7 + x)) = 105 ((35 + 5x) + x(7 + x)) = 105 (35 + 5x + (7 * x + x * x)) = 105 (35 + 5x + (7x + x2)) = 105 Combine like terms: 5x + 7x = 12x (35 + 12x + x2) = 105 Solving 35 + 12x + x2 = 105 Solving for variable 'x'. Reorder the terms: 35 + -105 + 12x + x2 = 105 + -105 Combine like terms: 35 + -105 = -70 -70 + 12x + x2 = 105 + -105 Combine like terms: 105 + -105 = 0 -70 + 12x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '70' to each side of the equation. -70 + 12x + 70 + x2 = 0 + 70 Reorder the terms: -70 + 70 + 12x + x2 = 0 + 70 Combine like terms: -70 + 70 = 0 0 + 12x + x2 = 0 + 70 12x + x2 = 0 + 70 Combine like terms: 0 + 70 = 70 12x + x2 = 70 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 = 70 + 36 Reorder the terms: 36 + 12x + x2 = 70 + 36 Combine like terms: 70 + 36 = 106 36 + 12x + x2 = 106 Factor a perfect square on the left side: (x + 6)(x + 6) = 106 Calculate the square root of the right side: 10.295630141 Break this problem into two subproblems by setting (x + 6) equal to 10.295630141 and -10.295630141.Subproblem 1
x + 6 = 10.295630141 Simplifying x + 6 = 10.295630141 Reorder the terms: 6 + x = 10.295630141 Solving 6 + x = 10.295630141 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 = 10.295630141 + -6 Combine like terms: 6 + -6 = 0 0 + x = 10.295630141 + -6 x = 10.295630141 + -6 Combine like terms: 10.295630141 + -6 = 4.295630141 x = 4.295630141 Simplifying x = 4.295630141Subproblem 2
x + 6 = -10.295630141 Simplifying x + 6 = -10.295630141 Reorder the terms: 6 + x = -10.295630141 Solving 6 + x = -10.295630141 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 = -10.295630141 + -6 Combine like terms: 6 + -6 = 0 0 + x = -10.295630141 + -6 x = -10.295630141 + -6 Combine like terms: -10.295630141 + -6 = -16.295630141 x = -16.295630141 Simplifying x = -16.295630141Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.295630141, -16.295630141}
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