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Simplifying 0 = 52x2 + -156x + -407 Reorder the terms: 0 = -407 + -156x + 52x2 Solving 0 = -407 + -156x + 52x2 Solving for variable 'x'. Combine like terms: 0 + 407 = 407 407 + 156x + -52x2 = -407 + -156x + 52x2 + 407 + 156x + -52x2 Reorder the terms: 407 + 156x + -52x2 = -407 + 407 + -156x + 156x + 52x2 + -52x2 Combine like terms: -407 + 407 = 0 407 + 156x + -52x2 = 0 + -156x + 156x + 52x2 + -52x2 407 + 156x + -52x2 = -156x + 156x + 52x2 + -52x2 Combine like terms: -156x + 156x = 0 407 + 156x + -52x2 = 0 + 52x2 + -52x2 407 + 156x + -52x2 = 52x2 + -52x2 Combine like terms: 52x2 + -52x2 = 0 407 + 156x + -52x2 = 0 Begin completing the square. Divide all terms by -52 the coefficient of the squared term: Divide each side by '-52'. -7.826923077 + -3x + x2 = 0 Move the constant term to the right: Add '7.826923077' to each side of the equation. -7.826923077 + -3x + 7.826923077 + x2 = 0 + 7.826923077 Reorder the terms: -7.826923077 + 7.826923077 + -3x + x2 = 0 + 7.826923077 Combine like terms: -7.826923077 + 7.826923077 = 0.000000000 0.000000000 + -3x + x2 = 0 + 7.826923077 -3x + x2 = 0 + 7.826923077 Combine like terms: 0 + 7.826923077 = 7.826923077 -3x + x2 = 7.826923077 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = 7.826923077 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 7.826923077 + 2.25 Combine like terms: 7.826923077 + 2.25 = 10.076923077 2.25 + -3x + x2 = 10.076923077 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 10.076923077 Calculate the square root of the right side: 3.174416966 Break this problem into two subproblems by setting (x + -1.5) equal to 3.174416966 and -3.174416966.Subproblem 1
x + -1.5 = 3.174416966 Simplifying x + -1.5 = 3.174416966 Reorder the terms: -1.5 + x = 3.174416966 Solving -1.5 + x = 3.174416966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 3.174416966 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 3.174416966 + 1.5 x = 3.174416966 + 1.5 Combine like terms: 3.174416966 + 1.5 = 4.674416966 x = 4.674416966 Simplifying x = 4.674416966Subproblem 2
x + -1.5 = -3.174416966 Simplifying x + -1.5 = -3.174416966 Reorder the terms: -1.5 + x = -3.174416966 Solving -1.5 + x = -3.174416966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -3.174416966 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -3.174416966 + 1.5 x = -3.174416966 + 1.5 Combine like terms: -3.174416966 + 1.5 = -1.674416966 x = -1.674416966 Simplifying x = -1.674416966Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.674416966, -1.674416966}
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