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Simplifying 1.411760 = 4x2 + 8x Reorder the terms: 1.411760 = 8x + 4x2 Solving 1.411760 = 8x + 4x2 Solving for variable 'x'. Reorder the terms: 1.411760 + -8x + -4x2 = 8x + -8x + 4x2 + -4x2 Combine like terms: 8x + -8x = 0 1.411760 + -8x + -4x2 = 0 + 4x2 + -4x2 1.411760 + -8x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 1.411760 + -8x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.35294 + 2x + x2 = 0 Move the constant term to the right: Add '0.35294' to each side of the equation. -0.35294 + 2x + 0.35294 + x2 = 0 + 0.35294 Reorder the terms: -0.35294 + 0.35294 + 2x + x2 = 0 + 0.35294 Combine like terms: -0.35294 + 0.35294 = 0.00000 0.00000 + 2x + x2 = 0 + 0.35294 2x + x2 = 0 + 0.35294 Combine like terms: 0 + 0.35294 = 0.35294 2x + x2 = 0.35294 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 0.35294 + 1 Reorder the terms: 1 + 2x + x2 = 0.35294 + 1 Combine like terms: 0.35294 + 1 = 1.35294 1 + 2x + x2 = 1.35294 Factor a perfect square on the left side: (x + 1)(x + 1) = 1.35294 Calculate the square root of the right side: 1.16315949 Break this problem into two subproblems by setting (x + 1) equal to 1.16315949 and -1.16315949.Subproblem 1
x + 1 = 1.16315949 Simplifying x + 1 = 1.16315949 Reorder the terms: 1 + x = 1.16315949 Solving 1 + x = 1.16315949 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 1.16315949 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.16315949 + -1 x = 1.16315949 + -1 Combine like terms: 1.16315949 + -1 = 0.16315949 x = 0.16315949 Simplifying x = 0.16315949Subproblem 2
x + 1 = -1.16315949 Simplifying x + 1 = -1.16315949 Reorder the terms: 1 + x = -1.16315949 Solving 1 + x = -1.16315949 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -1.16315949 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.16315949 + -1 x = -1.16315949 + -1 Combine like terms: -1.16315949 + -1 = -2.16315949 x = -2.16315949 Simplifying x = -2.16315949Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.16315949, -2.16315949}
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