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Simplifying 4x2 + 6.92820323x = -3 Reorder the terms: 6.92820323x + 4x2 = -3 Solving 6.92820323x + 4x2 = -3 Solving for variable 'x'. Reorder the terms: 3 + 6.92820323x + 4x2 = -3 + 3 Combine like terms: -3 + 3 = 0 3 + 6.92820323x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + 1.732050808x + x2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 1.732050808x + -0.75 + x2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 1.732050808x + x2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 1.732050808x + x2 = 0 + -0.75 1.732050808x + x2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 1.732050808x + x2 = -0.75 The x term is 1.732050808x. Take half its coefficient (0.866025404). Square it (0.7500000004) and add it to both sides. Add '0.7500000004' to each side of the equation. 1.732050808x + 0.7500000004 + x2 = -0.75 + 0.7500000004 Reorder the terms: 0.7500000004 + 1.732050808x + x2 = -0.75 + 0.7500000004 Combine like terms: -0.75 + 0.7500000004 = 0.0000000004 0.7500000004 + 1.732050808x + x2 = 0.0000000004 Factor a perfect square on the left side: (x + 0.866025404)(x + 0.866025404) = 0.0000000004 Calculate the square root of the right side: 0.00002 Break this problem into two subproblems by setting (x + 0.866025404) equal to 0.00002 and -0.00002.Subproblem 1
x + 0.866025404 = 0.00002 Simplifying x + 0.866025404 = 0.00002 Reorder the terms: 0.866025404 + x = 0.00002 Solving 0.866025404 + x = 0.00002 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.866025404' to each side of the equation. 0.866025404 + -0.866025404 + x = 0.00002 + -0.866025404 Combine like terms: 0.866025404 + -0.866025404 = 0.000000000 0.000000000 + x = 0.00002 + -0.866025404 x = 0.00002 + -0.866025404 Combine like terms: 0.00002 + -0.866025404 = -0.866005404 x = -0.866005404 Simplifying x = -0.866005404Subproblem 2
x + 0.866025404 = -0.00002 Simplifying x + 0.866025404 = -0.00002 Reorder the terms: 0.866025404 + x = -0.00002 Solving 0.866025404 + x = -0.00002 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.866025404' to each side of the equation. 0.866025404 + -0.866025404 + x = -0.00002 + -0.866025404 Combine like terms: 0.866025404 + -0.866025404 = 0.000000000 0.000000000 + x = -0.00002 + -0.866025404 x = -0.00002 + -0.866025404 Combine like terms: -0.00002 + -0.866025404 = -0.866045404 x = -0.866045404 Simplifying x = -0.866045404Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.866005404, -0.866045404}
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