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Simplifying (2x + 5)(x + -2) = 78 Reorder the terms: (5 + 2x)(x + -2) = 78 Reorder the terms: (5 + 2x)(-2 + x) = 78 Multiply (5 + 2x) * (-2 + x) (5(-2 + x) + 2x * (-2 + x)) = 78 ((-2 * 5 + x * 5) + 2x * (-2 + x)) = 78 ((-10 + 5x) + 2x * (-2 + x)) = 78 (-10 + 5x + (-2 * 2x + x * 2x)) = 78 (-10 + 5x + (-4x + 2x2)) = 78 Combine like terms: 5x + -4x = 1x (-10 + 1x + 2x2) = 78 Solving -10 + 1x + 2x2 = 78 Solving for variable 'x'. Reorder the terms: -10 + -78 + 1x + 2x2 = 78 + -78 Combine like terms: -10 + -78 = -88 -88 + 1x + 2x2 = 78 + -78 Combine like terms: 78 + -78 = 0 -88 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -44 + 0.5x + x2 = 0 Move the constant term to the right: Add '44' to each side of the equation. -44 + 0.5x + 44 + x2 = 0 + 44 Reorder the terms: -44 + 44 + 0.5x + x2 = 0 + 44 Combine like terms: -44 + 44 = 0 0 + 0.5x + x2 = 0 + 44 0.5x + x2 = 0 + 44 Combine like terms: 0 + 44 = 44 0.5x + x2 = 44 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 44 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 44 + 0.0625 Combine like terms: 44 + 0.0625 = 44.0625 0.0625 + 0.5x + x2 = 44.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 44.0625 Calculate the square root of the right side: 6.637959024 Break this problem into two subproblems by setting (x + 0.25) equal to 6.637959024 and -6.637959024.Subproblem 1
x + 0.25 = 6.637959024 Simplifying x + 0.25 = 6.637959024 Reorder the terms: 0.25 + x = 6.637959024 Solving 0.25 + x = 6.637959024 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 6.637959024 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 6.637959024 + -0.25 x = 6.637959024 + -0.25 Combine like terms: 6.637959024 + -0.25 = 6.387959024 x = 6.387959024 Simplifying x = 6.387959024Subproblem 2
x + 0.25 = -6.637959024 Simplifying x + 0.25 = -6.637959024 Reorder the terms: 0.25 + x = -6.637959024 Solving 0.25 + x = -6.637959024 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -6.637959024 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -6.637959024 + -0.25 x = -6.637959024 + -0.25 Combine like terms: -6.637959024 + -0.25 = -6.887959024 x = -6.887959024 Simplifying x = -6.887959024Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.387959024, -6.887959024}
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