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Simplifying (x + 7)(x + -4) = 10 Reorder the terms: (7 + x)(x + -4) = 10 Reorder the terms: (7 + x)(-4 + x) = 10 Multiply (7 + x) * (-4 + x) (7(-4 + x) + x(-4 + x)) = 10 ((-4 * 7 + x * 7) + x(-4 + x)) = 10 ((-28 + 7x) + x(-4 + x)) = 10 (-28 + 7x + (-4 * x + x * x)) = 10 (-28 + 7x + (-4x + x2)) = 10 Combine like terms: 7x + -4x = 3x (-28 + 3x + x2) = 10 Solving -28 + 3x + x2 = 10 Solving for variable 'x'. Reorder the terms: -28 + -10 + 3x + x2 = 10 + -10 Combine like terms: -28 + -10 = -38 -38 + 3x + x2 = 10 + -10 Combine like terms: 10 + -10 = 0 -38 + 3x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '38' to each side of the equation. -38 + 3x + 38 + x2 = 0 + 38 Reorder the terms: -38 + 38 + 3x + x2 = 0 + 38 Combine like terms: -38 + 38 = 0 0 + 3x + x2 = 0 + 38 3x + x2 = 0 + 38 Combine like terms: 0 + 38 = 38 3x + x2 = 38 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 38 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 38 + 2.25 Combine like terms: 38 + 2.25 = 40.25 2.25 + 3x + x2 = 40.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 40.25 Calculate the square root of the right side: 6.34428877 Break this problem into two subproblems by setting (x + 1.5) equal to 6.34428877 and -6.34428877.Subproblem 1
x + 1.5 = 6.34428877 Simplifying x + 1.5 = 6.34428877 Reorder the terms: 1.5 + x = 6.34428877 Solving 1.5 + x = 6.34428877 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 6.34428877 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 6.34428877 + -1.5 x = 6.34428877 + -1.5 Combine like terms: 6.34428877 + -1.5 = 4.84428877 x = 4.84428877 Simplifying x = 4.84428877Subproblem 2
x + 1.5 = -6.34428877 Simplifying x + 1.5 = -6.34428877 Reorder the terms: 1.5 + x = -6.34428877 Solving 1.5 + x = -6.34428877 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -6.34428877 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -6.34428877 + -1.5 x = -6.34428877 + -1.5 Combine like terms: -6.34428877 + -1.5 = -7.84428877 x = -7.84428877 Simplifying x = -7.84428877Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.84428877, -7.84428877}
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