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Simplifying 8x2 + 19x + 3 = 0 Reorder the terms: 3 + 19x + 8x2 = 0 Solving 3 + 19x + 8x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.375 + 2.375x + x2 = 0 Move the constant term to the right: Add '-0.375' to each side of the equation. 0.375 + 2.375x + -0.375 + x2 = 0 + -0.375 Reorder the terms: 0.375 + -0.375 + 2.375x + x2 = 0 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + 2.375x + x2 = 0 + -0.375 2.375x + x2 = 0 + -0.375 Combine like terms: 0 + -0.375 = -0.375 2.375x + x2 = -0.375 The x term is 2.375x. Take half its coefficient (1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. 2.375x + 1.41015625 + x2 = -0.375 + 1.41015625 Reorder the terms: 1.41015625 + 2.375x + x2 = -0.375 + 1.41015625 Combine like terms: -0.375 + 1.41015625 = 1.03515625 1.41015625 + 2.375x + x2 = 1.03515625 Factor a perfect square on the left side: (x + 1.1875)(x + 1.1875) = 1.03515625 Calculate the square root of the right side: 1.017426287 Break this problem into two subproblems by setting (x + 1.1875) equal to 1.017426287 and -1.017426287.Subproblem 1
x + 1.1875 = 1.017426287 Simplifying x + 1.1875 = 1.017426287 Reorder the terms: 1.1875 + x = 1.017426287 Solving 1.1875 + x = 1.017426287 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.1875' to each side of the equation. 1.1875 + -1.1875 + x = 1.017426287 + -1.1875 Combine like terms: 1.1875 + -1.1875 = 0.0000 0.0000 + x = 1.017426287 + -1.1875 x = 1.017426287 + -1.1875 Combine like terms: 1.017426287 + -1.1875 = -0.170073713 x = -0.170073713 Simplifying x = -0.170073713Subproblem 2
x + 1.1875 = -1.017426287 Simplifying x + 1.1875 = -1.017426287 Reorder the terms: 1.1875 + x = -1.017426287 Solving 1.1875 + x = -1.017426287 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.1875' to each side of the equation. 1.1875 + -1.1875 + x = -1.017426287 + -1.1875 Combine like terms: 1.1875 + -1.1875 = 0.0000 0.0000 + x = -1.017426287 + -1.1875 x = -1.017426287 + -1.1875 Combine like terms: -1.017426287 + -1.1875 = -2.204926287 x = -2.204926287 Simplifying x = -2.204926287Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.170073713, -2.204926287}
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