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Simplifying 53x2 + -128x + 35 = 0 Reorder the terms: 35 + -128x + 53x2 = 0 Solving 35 + -128x + 53x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 53 the coefficient of the squared term: Divide each side by '53'. 0.6603773585 + -2.41509434x + x2 = 0 Move the constant term to the right: Add '-0.6603773585' to each side of the equation. 0.6603773585 + -2.41509434x + -0.6603773585 + x2 = 0 + -0.6603773585 Reorder the terms: 0.6603773585 + -0.6603773585 + -2.41509434x + x2 = 0 + -0.6603773585 Combine like terms: 0.6603773585 + -0.6603773585 = 0.0000000000 0.0000000000 + -2.41509434x + x2 = 0 + -0.6603773585 -2.41509434x + x2 = 0 + -0.6603773585 Combine like terms: 0 + -0.6603773585 = -0.6603773585 -2.41509434x + x2 = -0.6603773585 The x term is -2.41509434x. Take half its coefficient (-1.20754717). Square it (1.458170168) and add it to both sides. Add '1.458170168' to each side of the equation. -2.41509434x + 1.458170168 + x2 = -0.6603773585 + 1.458170168 Reorder the terms: 1.458170168 + -2.41509434x + x2 = -0.6603773585 + 1.458170168 Combine like terms: -0.6603773585 + 1.458170168 = 0.7977928095 1.458170168 + -2.41509434x + x2 = 0.7977928095 Factor a perfect square on the left side: (x + -1.20754717)(x + -1.20754717) = 0.7977928095 Calculate the square root of the right side: 0.893192482 Break this problem into two subproblems by setting (x + -1.20754717) equal to 0.893192482 and -0.893192482.Subproblem 1
x + -1.20754717 = 0.893192482 Simplifying x + -1.20754717 = 0.893192482 Reorder the terms: -1.20754717 + x = 0.893192482 Solving -1.20754717 + x = 0.893192482 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.20754717' to each side of the equation. -1.20754717 + 1.20754717 + x = 0.893192482 + 1.20754717 Combine like terms: -1.20754717 + 1.20754717 = 0.00000000 0.00000000 + x = 0.893192482 + 1.20754717 x = 0.893192482 + 1.20754717 Combine like terms: 0.893192482 + 1.20754717 = 2.100739652 x = 2.100739652 Simplifying x = 2.100739652Subproblem 2
x + -1.20754717 = -0.893192482 Simplifying x + -1.20754717 = -0.893192482 Reorder the terms: -1.20754717 + x = -0.893192482 Solving -1.20754717 + x = -0.893192482 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.20754717' to each side of the equation. -1.20754717 + 1.20754717 + x = -0.893192482 + 1.20754717 Combine like terms: -1.20754717 + 1.20754717 = 0.00000000 0.00000000 + x = -0.893192482 + 1.20754717 x = -0.893192482 + 1.20754717 Combine like terms: -0.893192482 + 1.20754717 = 0.314354688 x = 0.314354688 Simplifying x = 0.314354688Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.100739652, 0.314354688}
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