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Simplifying 3(5 + -7x) + 2(1 + 4x)(7 + -5x) = 0 (5 * 3 + -7x * 3) + 2(1 + 4x)(7 + -5x) = 0 (15 + -21x) + 2(1 + 4x)(7 + -5x) = 0 Multiply (1 + 4x) * (7 + -5x) 15 + -21x + 2(1(7 + -5x) + 4x * (7 + -5x)) = 0 15 + -21x + 2((7 * 1 + -5x * 1) + 4x * (7 + -5x)) = 0 15 + -21x + 2((7 + -5x) + 4x * (7 + -5x)) = 0 15 + -21x + 2(7 + -5x + (7 * 4x + -5x * 4x)) = 0 15 + -21x + 2(7 + -5x + (28x + -20x2)) = 0 Combine like terms: -5x + 28x = 23x 15 + -21x + 2(7 + 23x + -20x2) = 0 15 + -21x + (7 * 2 + 23x * 2 + -20x2 * 2) = 0 15 + -21x + (14 + 46x + -40x2) = 0 Reorder the terms: 15 + 14 + -21x + 46x + -40x2 = 0 Combine like terms: 15 + 14 = 29 29 + -21x + 46x + -40x2 = 0 Combine like terms: -21x + 46x = 25x 29 + 25x + -40x2 = 0 Solving 29 + 25x + -40x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -40 the coefficient of the squared term: Divide each side by '-40'. -0.725 + -0.625x + x2 = 0.0 Move the constant term to the right: Add '0.725' to each side of the equation. -0.725 + -0.625x + 0.725 + x2 = 0.0 + 0.725 Reorder the terms: -0.725 + 0.725 + -0.625x + x2 = 0.0 + 0.725 Combine like terms: -0.725 + 0.725 = 0.000 0.000 + -0.625x + x2 = 0.0 + 0.725 -0.625x + x2 = 0.0 + 0.725 Combine like terms: 0.0 + 0.725 = 0.725 -0.625x + x2 = 0.725 The x term is -0.625x. Take half its coefficient (-0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. -0.625x + 0.09765625 + x2 = 0.725 + 0.09765625 Reorder the terms: 0.09765625 + -0.625x + x2 = 0.725 + 0.09765625 Combine like terms: 0.725 + 0.09765625 = 0.82265625 0.09765625 + -0.625x + x2 = 0.82265625 Factor a perfect square on the left side: (x + -0.3125)(x + -0.3125) = 0.82265625 Calculate the square root of the right side: 0.907003997 Break this problem into two subproblems by setting (x + -0.3125) equal to 0.907003997 and -0.907003997.Subproblem 1
x + -0.3125 = 0.907003997 Simplifying x + -0.3125 = 0.907003997 Reorder the terms: -0.3125 + x = 0.907003997 Solving -0.3125 + x = 0.907003997 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + x = 0.907003997 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + x = 0.907003997 + 0.3125 x = 0.907003997 + 0.3125 Combine like terms: 0.907003997 + 0.3125 = 1.219503997 x = 1.219503997 Simplifying x = 1.219503997Subproblem 2
x + -0.3125 = -0.907003997 Simplifying x + -0.3125 = -0.907003997 Reorder the terms: -0.3125 + x = -0.907003997 Solving -0.3125 + x = -0.907003997 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + x = -0.907003997 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + x = -0.907003997 + 0.3125 x = -0.907003997 + 0.3125 Combine like terms: -0.907003997 + 0.3125 = -0.594503997 x = -0.594503997 Simplifying x = -0.594503997Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.219503997, -0.594503997}
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