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Simplifying 23.4918 = 14x + -2x2 Solving 23.4918 = 14x + -2x2 Solving for variable 'x'. Reorder the terms: 23.4918 + -14x + 2x2 = 14x + -14x + -2x2 + 2x2 Combine like terms: 14x + -14x = 0 23.4918 + -14x + 2x2 = 0 + -2x2 + 2x2 23.4918 + -14x + 2x2 = -2x2 + 2x2 Combine like terms: -2x2 + 2x2 = 0 23.4918 + -14x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 11.7459 + -7x + x2 = 0 Move the constant term to the right: Add '-11.7459' to each side of the equation. 11.7459 + -7x + -11.7459 + x2 = 0 + -11.7459 Reorder the terms: 11.7459 + -11.7459 + -7x + x2 = 0 + -11.7459 Combine like terms: 11.7459 + -11.7459 = 0.0000 0.0000 + -7x + x2 = 0 + -11.7459 -7x + x2 = 0 + -11.7459 Combine like terms: 0 + -11.7459 = -11.7459 -7x + x2 = -11.7459 The x term is -7x. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7x + 12.25 + x2 = -11.7459 + 12.25 Reorder the terms: 12.25 + -7x + x2 = -11.7459 + 12.25 Combine like terms: -11.7459 + 12.25 = 0.5041 12.25 + -7x + x2 = 0.5041 Factor a perfect square on the left side: (x + -3.5)(x + -3.5) = 0.5041 Calculate the square root of the right side: 0.71 Break this problem into two subproblems by setting (x + -3.5) equal to 0.71 and -0.71.Subproblem 1
x + -3.5 = 0.71 Simplifying x + -3.5 = 0.71 Reorder the terms: -3.5 + x = 0.71 Solving -3.5 + x = 0.71 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 0.71 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 0.71 + 3.5 x = 0.71 + 3.5 Combine like terms: 0.71 + 3.5 = 4.21 x = 4.21 Simplifying x = 4.21Subproblem 2
x + -3.5 = -0.71 Simplifying x + -3.5 = -0.71 Reorder the terms: -3.5 + x = -0.71 Solving -3.5 + x = -0.71 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -0.71 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -0.71 + 3.5 x = -0.71 + 3.5 Combine like terms: -0.71 + 3.5 = 2.79 x = 2.79 Simplifying x = 2.79Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.21, 2.79}
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