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Simplifying 7x(1 + -4x) + 5(1 + 2x) = 0 (1 * 7x + -4x * 7x) + 5(1 + 2x) = 0 (7x + -28x2) + 5(1 + 2x) = 0 7x + -28x2 + (1 * 5 + 2x * 5) = 0 7x + -28x2 + (5 + 10x) = 0 Reorder the terms: 5 + 7x + 10x + -28x2 = 0 Combine like terms: 7x + 10x = 17x 5 + 17x + -28x2 = 0 Solving 5 + 17x + -28x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -28 the coefficient of the squared term: Divide each side by '-28'. -0.1785714286 + -0.6071428571x + x2 = 0 Move the constant term to the right: Add '0.1785714286' to each side of the equation. -0.1785714286 + -0.6071428571x + 0.1785714286 + x2 = 0 + 0.1785714286 Reorder the terms: -0.1785714286 + 0.1785714286 + -0.6071428571x + x2 = 0 + 0.1785714286 Combine like terms: -0.1785714286 + 0.1785714286 = 0.0000000000 0.0000000000 + -0.6071428571x + x2 = 0 + 0.1785714286 -0.6071428571x + x2 = 0 + 0.1785714286 Combine like terms: 0 + 0.1785714286 = 0.1785714286 -0.6071428571x + x2 = 0.1785714286 The x term is -0.6071428571x. Take half its coefficient (-0.3035714286). Square it (0.09215561226) and add it to both sides. Add '0.09215561226' to each side of the equation. -0.6071428571x + 0.09215561226 + x2 = 0.1785714286 + 0.09215561226 Reorder the terms: 0.09215561226 + -0.6071428571x + x2 = 0.1785714286 + 0.09215561226 Combine like terms: 0.1785714286 + 0.09215561226 = 0.27072704086 0.09215561226 + -0.6071428571x + x2 = 0.27072704086 Factor a perfect square on the left side: (x + -0.3035714286)(x + -0.3035714286) = 0.27072704086 Calculate the square root of the right side: 0.520314367 Break this problem into two subproblems by setting (x + -0.3035714286) equal to 0.520314367 and -0.520314367.Subproblem 1
x + -0.3035714286 = 0.520314367 Simplifying x + -0.3035714286 = 0.520314367 Reorder the terms: -0.3035714286 + x = 0.520314367 Solving -0.3035714286 + x = 0.520314367 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3035714286' to each side of the equation. -0.3035714286 + 0.3035714286 + x = 0.520314367 + 0.3035714286 Combine like terms: -0.3035714286 + 0.3035714286 = 0.0000000000 0.0000000000 + x = 0.520314367 + 0.3035714286 x = 0.520314367 + 0.3035714286 Combine like terms: 0.520314367 + 0.3035714286 = 0.8238857956 x = 0.8238857956 Simplifying x = 0.8238857956Subproblem 2
x + -0.3035714286 = -0.520314367 Simplifying x + -0.3035714286 = -0.520314367 Reorder the terms: -0.3035714286 + x = -0.520314367 Solving -0.3035714286 + x = -0.520314367 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3035714286' to each side of the equation. -0.3035714286 + 0.3035714286 + x = -0.520314367 + 0.3035714286 Combine like terms: -0.3035714286 + 0.3035714286 = 0.0000000000 0.0000000000 + x = -0.520314367 + 0.3035714286 x = -0.520314367 + 0.3035714286 Combine like terms: -0.520314367 + 0.3035714286 = -0.2167429384 x = -0.2167429384 Simplifying x = -0.2167429384Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.8238857956, -0.2167429384}
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