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Simplifying [(7x + 4)(-8x + -8)] + [(8x + 9)(5x + 6)] = 0 Reorder the terms: [(4 + 7x)(-8x + -8)] + [(8x + 9)(5x + 6)] = 0 Reorder the terms: [(4 + 7x)(-8 + -8x)] + [(8x + 9)(5x + 6)] = 0 Multiply (4 + 7x) * (-8 + -8x) [(4(-8 + -8x) + 7x * (-8 + -8x))] + [(8x + 9)(5x + 6)] = 0 [((-8 * 4 + -8x * 4) + 7x * (-8 + -8x))] + [(8x + 9)(5x + 6)] = 0 [((-32 + -32x) + 7x * (-8 + -8x))] + [(8x + 9)(5x + 6)] = 0 [(-32 + -32x + (-8 * 7x + -8x * 7x))] + [(8x + 9)(5x + 6)] = 0 [(-32 + -32x + (-56x + -56x2))] + [(8x + 9)(5x + 6)] = 0 Combine like terms: -32x + -56x = -88x [(-32 + -88x + -56x2)] + [(8x + 9)(5x + 6)] = 0 [-32 + -88x + -56x2] + [(8x + 9)(5x + 6)] = 0 Remove brackets around [-32 + -88x + -56x2] -32 + -88x + -56x2 + [(8x + 9)(5x + 6)] = 0 Reorder the terms: -32 + -88x + -56x2 + [(9 + 8x)(5x + 6)] = 0 Reorder the terms: -32 + -88x + -56x2 + [(9 + 8x)(6 + 5x)] = 0 Multiply (9 + 8x) * (6 + 5x) -32 + -88x + -56x2 + [(9(6 + 5x) + 8x * (6 + 5x))] = 0 -32 + -88x + -56x2 + [((6 * 9 + 5x * 9) + 8x * (6 + 5x))] = 0 -32 + -88x + -56x2 + [((54 + 45x) + 8x * (6 + 5x))] = 0 -32 + -88x + -56x2 + [(54 + 45x + (6 * 8x + 5x * 8x))] = 0 -32 + -88x + -56x2 + [(54 + 45x + (48x + 40x2))] = 0 Combine like terms: 45x + 48x = 93x -32 + -88x + -56x2 + [(54 + 93x + 40x2)] = 0 -32 + -88x + -56x2 + [54 + 93x + 40x2] = 0 Remove brackets around [54 + 93x + 40x2] -32 + -88x + -56x2 + 54 + 93x + 40x2 = 0 Reorder the terms: -32 + 54 + -88x + 93x + -56x2 + 40x2 = 0 Combine like terms: -32 + 54 = 22 22 + -88x + 93x + -56x2 + 40x2 = 0 Combine like terms: -88x + 93x = 5x 22 + 5x + -56x2 + 40x2 = 0 Combine like terms: -56x2 + 40x2 = -16x2 22 + 5x + -16x2 = 0 Solving 22 + 5x + -16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -1.375 + -0.3125x + x2 = 0 Move the constant term to the right: Add '1.375' to each side of the equation. -1.375 + -0.3125x + 1.375 + x2 = 0 + 1.375 Reorder the terms: -1.375 + 1.375 + -0.3125x + x2 = 0 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + -0.3125x + x2 = 0 + 1.375 -0.3125x + x2 = 0 + 1.375 Combine like terms: 0 + 1.375 = 1.375 -0.3125x + x2 = 1.375 The x term is -0.3125x. Take half its coefficient (-0.15625). Square it (0.0244140625) and add it to both sides. Add '0.0244140625' to each side of the equation. -0.3125x + 0.0244140625 + x2 = 1.375 + 0.0244140625 Reorder the terms: 0.0244140625 + -0.3125x + x2 = 1.375 + 0.0244140625 Combine like terms: 1.375 + 0.0244140625 = 1.3994140625 0.0244140625 + -0.3125x + x2 = 1.3994140625 Factor a perfect square on the left side: (x + -0.15625)(x + -0.15625) = 1.3994140625 Calculate the square root of the right side: 1.182968327 Break this problem into two subproblems by setting (x + -0.15625) equal to 1.182968327 and -1.182968327.Subproblem 1
x + -0.15625 = 1.182968327 Simplifying x + -0.15625 = 1.182968327 Reorder the terms: -0.15625 + x = 1.182968327 Solving -0.15625 + x = 1.182968327 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.15625' to each side of the equation. -0.15625 + 0.15625 + x = 1.182968327 + 0.15625 Combine like terms: -0.15625 + 0.15625 = 0.00000 0.00000 + x = 1.182968327 + 0.15625 x = 1.182968327 + 0.15625 Combine like terms: 1.182968327 + 0.15625 = 1.339218327 x = 1.339218327 Simplifying x = 1.339218327Subproblem 2
x + -0.15625 = -1.182968327 Simplifying x + -0.15625 = -1.182968327 Reorder the terms: -0.15625 + x = -1.182968327 Solving -0.15625 + x = -1.182968327 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.15625' to each side of the equation. -0.15625 + 0.15625 + x = -1.182968327 + 0.15625 Combine like terms: -0.15625 + 0.15625 = 0.00000 0.00000 + x = -1.182968327 + 0.15625 x = -1.182968327 + 0.15625 Combine like terms: -1.182968327 + 0.15625 = -1.026718327 x = -1.026718327 Simplifying x = -1.026718327Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.339218327, -1.026718327}
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