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Simplifying 5(3 + -4x2) + -8(3 + -4x)(5x + -1) = 0 (3 * 5 + -4x2 * 5) + -8(3 + -4x)(5x + -1) = 0 (15 + -20x2) + -8(3 + -4x)(5x + -1) = 0 Reorder the terms: 15 + -20x2 + -8(3 + -4x)(-1 + 5x) = 0 Multiply (3 + -4x) * (-1 + 5x) 15 + -20x2 + -8(3(-1 + 5x) + -4x * (-1 + 5x)) = 0 15 + -20x2 + -8((-1 * 3 + 5x * 3) + -4x * (-1 + 5x)) = 0 15 + -20x2 + -8((-3 + 15x) + -4x * (-1 + 5x)) = 0 15 + -20x2 + -8(-3 + 15x + (-1 * -4x + 5x * -4x)) = 0 15 + -20x2 + -8(-3 + 15x + (4x + -20x2)) = 0 Combine like terms: 15x + 4x = 19x 15 + -20x2 + -8(-3 + 19x + -20x2) = 0 15 + -20x2 + (-3 * -8 + 19x * -8 + -20x2 * -8) = 0 15 + -20x2 + (24 + -152x + 160x2) = 0 Reorder the terms: 15 + 24 + -152x + -20x2 + 160x2 = 0 Combine like terms: 15 + 24 = 39 39 + -152x + -20x2 + 160x2 = 0 Combine like terms: -20x2 + 160x2 = 140x2 39 + -152x + 140x2 = 0 Solving 39 + -152x + 140x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 140 the coefficient of the squared term: Divide each side by '140'. 0.2785714286 + -1.085714286x + x2 = 0.0 Move the constant term to the right: Add '-0.2785714286' to each side of the equation. 0.2785714286 + -1.085714286x + -0.2785714286 + x2 = 0.0 + -0.2785714286 Reorder the terms: 0.2785714286 + -0.2785714286 + -1.085714286x + x2 = 0.0 + -0.2785714286 Combine like terms: 0.2785714286 + -0.2785714286 = 0.0000000000 0.0000000000 + -1.085714286x + x2 = 0.0 + -0.2785714286 -1.085714286x + x2 = 0.0 + -0.2785714286 Combine like terms: 0.0 + -0.2785714286 = -0.2785714286 -1.085714286x + x2 = -0.2785714286 The x term is -1.085714286x. Take half its coefficient (-0.542857143). Square it (0.2946938777) and add it to both sides. Add '0.2946938777' to each side of the equation. -1.085714286x + 0.2946938777 + x2 = -0.2785714286 + 0.2946938777 Reorder the terms: 0.2946938777 + -1.085714286x + x2 = -0.2785714286 + 0.2946938777 Combine like terms: -0.2785714286 + 0.2946938777 = 0.0161224491 0.2946938777 + -1.085714286x + x2 = 0.0161224491 Factor a perfect square on the left side: (x + -0.542857143)(x + -0.542857143) = 0.0161224491 Calculate the square root of the right side: 0.126974206 Break this problem into two subproblems by setting (x + -0.542857143) equal to 0.126974206 and -0.126974206.Subproblem 1
x + -0.542857143 = 0.126974206 Simplifying x + -0.542857143 = 0.126974206 Reorder the terms: -0.542857143 + x = 0.126974206 Solving -0.542857143 + x = 0.126974206 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.542857143' to each side of the equation. -0.542857143 + 0.542857143 + x = 0.126974206 + 0.542857143 Combine like terms: -0.542857143 + 0.542857143 = 0.000000000 0.000000000 + x = 0.126974206 + 0.542857143 x = 0.126974206 + 0.542857143 Combine like terms: 0.126974206 + 0.542857143 = 0.669831349 x = 0.669831349 Simplifying x = 0.669831349Subproblem 2
x + -0.542857143 = -0.126974206 Simplifying x + -0.542857143 = -0.126974206 Reorder the terms: -0.542857143 + x = -0.126974206 Solving -0.542857143 + x = -0.126974206 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.542857143' to each side of the equation. -0.542857143 + 0.542857143 + x = -0.126974206 + 0.542857143 Combine like terms: -0.542857143 + 0.542857143 = 0.000000000 0.000000000 + x = -0.126974206 + 0.542857143 x = -0.126974206 + 0.542857143 Combine like terms: -0.126974206 + 0.542857143 = 0.415882937 x = 0.415882937 Simplifying x = 0.415882937Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.669831349, 0.415882937}
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