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Simplifying (x + 5)(x + -7) = 14 Reorder the terms: (5 + x)(x + -7) = 14 Reorder the terms: (5 + x)(-7 + x) = 14 Multiply (5 + x) * (-7 + x) (5(-7 + x) + x(-7 + x)) = 14 ((-7 * 5 + x * 5) + x(-7 + x)) = 14 ((-35 + 5x) + x(-7 + x)) = 14 (-35 + 5x + (-7 * x + x * x)) = 14 (-35 + 5x + (-7x + x2)) = 14 Combine like terms: 5x + -7x = -2x (-35 + -2x + x2) = 14 Solving -35 + -2x + x2 = 14 Solving for variable 'x'. Reorder the terms: -35 + -14 + -2x + x2 = 14 + -14 Combine like terms: -35 + -14 = -49 -49 + -2x + x2 = 14 + -14 Combine like terms: 14 + -14 = 0 -49 + -2x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '49' to each side of the equation. -49 + -2x + 49 + x2 = 0 + 49 Reorder the terms: -49 + 49 + -2x + x2 = 0 + 49 Combine like terms: -49 + 49 = 0 0 + -2x + x2 = 0 + 49 -2x + x2 = 0 + 49 Combine like terms: 0 + 49 = 49 -2x + x2 = 49 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = 49 + 1 Reorder the terms: 1 + -2x + x2 = 49 + 1 Combine like terms: 49 + 1 = 50 1 + -2x + x2 = 50 Factor a perfect square on the left side: (x + -1)(x + -1) = 50 Calculate the square root of the right side: 7.071067812 Break this problem into two subproblems by setting (x + -1) equal to 7.071067812 and -7.071067812.Subproblem 1
x + -1 = 7.071067812 Simplifying x + -1 = 7.071067812 Reorder the terms: -1 + x = 7.071067812 Solving -1 + x = 7.071067812 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 7.071067812 + 1 Combine like terms: -1 + 1 = 0 0 + x = 7.071067812 + 1 x = 7.071067812 + 1 Combine like terms: 7.071067812 + 1 = 8.071067812 x = 8.071067812 Simplifying x = 8.071067812Subproblem 2
x + -1 = -7.071067812 Simplifying x + -1 = -7.071067812 Reorder the terms: -1 + x = -7.071067812 Solving -1 + x = -7.071067812 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -7.071067812 + 1 Combine like terms: -1 + 1 = 0 0 + x = -7.071067812 + 1 x = -7.071067812 + 1 Combine like terms: -7.071067812 + 1 = -6.071067812 x = -6.071067812 Simplifying x = -6.071067812Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.071067812, -6.071067812}
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