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Simplifying 5(2 + -1x) * 7x = -3(x + 5) Reorder the terms for easier multiplication: 5 * 7x(2 + -1x) = -3(x + 5) Multiply 5 * 7 35x(2 + -1x) = -3(x + 5) (2 * 35x + -1x * 35x) = -3(x + 5) (70x + -35x2) = -3(x + 5) Reorder the terms: 70x + -35x2 = -3(5 + x) 70x + -35x2 = (5 * -3 + x * -3) 70x + -35x2 = (-15 + -3x) Solving 70x + -35x2 = -15 + -3x Solving for variable 'x'. Reorder the terms: 15 + 70x + 3x + -35x2 = -15 + -3x + 15 + 3x Combine like terms: 70x + 3x = 73x 15 + 73x + -35x2 = -15 + -3x + 15 + 3x Reorder the terms: 15 + 73x + -35x2 = -15 + 15 + -3x + 3x Combine like terms: -15 + 15 = 0 15 + 73x + -35x2 = 0 + -3x + 3x 15 + 73x + -35x2 = -3x + 3x Combine like terms: -3x + 3x = 0 15 + 73x + -35x2 = 0 Begin completing the square. Divide all terms by -35 the coefficient of the squared term: Divide each side by '-35'. -0.4285714286 + -2.085714286x + x2 = 0 Move the constant term to the right: Add '0.4285714286' to each side of the equation. -0.4285714286 + -2.085714286x + 0.4285714286 + x2 = 0 + 0.4285714286 Reorder the terms: -0.4285714286 + 0.4285714286 + -2.085714286x + x2 = 0 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + -2.085714286x + x2 = 0 + 0.4285714286 -2.085714286x + x2 = 0 + 0.4285714286 Combine like terms: 0 + 0.4285714286 = 0.4285714286 -2.085714286x + x2 = 0.4285714286 The x term is -2.085714286x. Take half its coefficient (-1.042857143). Square it (1.087551021) and add it to both sides. Add '1.087551021' to each side of the equation. -2.085714286x + 1.087551021 + x2 = 0.4285714286 + 1.087551021 Reorder the terms: 1.087551021 + -2.085714286x + x2 = 0.4285714286 + 1.087551021 Combine like terms: 0.4285714286 + 1.087551021 = 1.5161224496 1.087551021 + -2.085714286x + x2 = 1.5161224496 Factor a perfect square on the left side: (x + -1.042857143)(x + -1.042857143) = 1.5161224496 Calculate the square root of the right side: 1.231309242 Break this problem into two subproblems by setting (x + -1.042857143) equal to 1.231309242 and -1.231309242.Subproblem 1
x + -1.042857143 = 1.231309242 Simplifying x + -1.042857143 = 1.231309242 Reorder the terms: -1.042857143 + x = 1.231309242 Solving -1.042857143 + x = 1.231309242 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.042857143' to each side of the equation. -1.042857143 + 1.042857143 + x = 1.231309242 + 1.042857143 Combine like terms: -1.042857143 + 1.042857143 = 0.000000000 0.000000000 + x = 1.231309242 + 1.042857143 x = 1.231309242 + 1.042857143 Combine like terms: 1.231309242 + 1.042857143 = 2.274166385 x = 2.274166385 Simplifying x = 2.274166385Subproblem 2
x + -1.042857143 = -1.231309242 Simplifying x + -1.042857143 = -1.231309242 Reorder the terms: -1.042857143 + x = -1.231309242 Solving -1.042857143 + x = -1.231309242 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.042857143' to each side of the equation. -1.042857143 + 1.042857143 + x = -1.231309242 + 1.042857143 Combine like terms: -1.042857143 + 1.042857143 = 0.000000000 0.000000000 + x = -1.231309242 + 1.042857143 x = -1.231309242 + 1.042857143 Combine like terms: -1.231309242 + 1.042857143 = -0.188452099 x = -0.188452099 Simplifying x = -0.188452099Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.274166385, -0.188452099}
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