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Simplifying 9x2 + 14x + -19 = 0 Reorder the terms: -19 + 14x + 9x2 = 0 Solving -19 + 14x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.111111111 + 1.555555556x + x2 = 0 Move the constant term to the right: Add '2.111111111' to each side of the equation. -2.111111111 + 1.555555556x + 2.111111111 + x2 = 0 + 2.111111111 Reorder the terms: -2.111111111 + 2.111111111 + 1.555555556x + x2 = 0 + 2.111111111 Combine like terms: -2.111111111 + 2.111111111 = 0.000000000 0.000000000 + 1.555555556x + x2 = 0 + 2.111111111 1.555555556x + x2 = 0 + 2.111111111 Combine like terms: 0 + 2.111111111 = 2.111111111 1.555555556x + x2 = 2.111111111 The x term is 1.555555556x. Take half its coefficient (0.777777778). Square it (0.6049382720) and add it to both sides. Add '0.6049382720' to each side of the equation. 1.555555556x + 0.6049382720 + x2 = 2.111111111 + 0.6049382720 Reorder the terms: 0.6049382720 + 1.555555556x + x2 = 2.111111111 + 0.6049382720 Combine like terms: 2.111111111 + 0.6049382720 = 2.716049383 0.6049382720 + 1.555555556x + x2 = 2.716049383 Factor a perfect square on the left side: (x + 0.777777778)(x + 0.777777778) = 2.716049383 Calculate the square root of the right side: 1.648044108 Break this problem into two subproblems by setting (x + 0.777777778) equal to 1.648044108 and -1.648044108.Subproblem 1
x + 0.777777778 = 1.648044108 Simplifying x + 0.777777778 = 1.648044108 Reorder the terms: 0.777777778 + x = 1.648044108 Solving 0.777777778 + x = 1.648044108 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + x = 1.648044108 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + x = 1.648044108 + -0.777777778 x = 1.648044108 + -0.777777778 Combine like terms: 1.648044108 + -0.777777778 = 0.87026633 x = 0.87026633 Simplifying x = 0.87026633Subproblem 2
x + 0.777777778 = -1.648044108 Simplifying x + 0.777777778 = -1.648044108 Reorder the terms: 0.777777778 + x = -1.648044108 Solving 0.777777778 + x = -1.648044108 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + x = -1.648044108 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + x = -1.648044108 + -0.777777778 x = -1.648044108 + -0.777777778 Combine like terms: -1.648044108 + -0.777777778 = -2.425821886 x = -2.425821886 Simplifying x = -2.425821886Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.87026633, -2.425821886}
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