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Simplifying 25 = -16x2 + 52x + 19 Reorder the terms: 25 = 19 + 52x + -16x2 Solving 25 = 19 + 52x + -16x2 Solving for variable 'x'. Combine like terms: 25 + -19 = 6 6 + -52x + 16x2 = 19 + 52x + -16x2 + -19 + -52x + 16x2 Reorder the terms: 6 + -52x + 16x2 = 19 + -19 + 52x + -52x + -16x2 + 16x2 Combine like terms: 19 + -19 = 0 6 + -52x + 16x2 = 0 + 52x + -52x + -16x2 + 16x2 6 + -52x + 16x2 = 52x + -52x + -16x2 + 16x2 Combine like terms: 52x + -52x = 0 6 + -52x + 16x2 = 0 + -16x2 + 16x2 6 + -52x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 6 + -52x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(3 + -26x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(3 + -26x + 8x2)' equal to zero and attempt to solve: Simplifying 3 + -26x + 8x2 = 0 Solving 3 + -26x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.375 + -3.25x + x2 = 0 Move the constant term to the right: Add '-0.375' to each side of the equation. 0.375 + -3.25x + -0.375 + x2 = 0 + -0.375 Reorder the terms: 0.375 + -0.375 + -3.25x + x2 = 0 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + -3.25x + x2 = 0 + -0.375 -3.25x + x2 = 0 + -0.375 Combine like terms: 0 + -0.375 = -0.375 -3.25x + x2 = -0.375 The x term is -3.25x. Take half its coefficient (-1.625). Square it (2.640625) and add it to both sides. Add '2.640625' to each side of the equation. -3.25x + 2.640625 + x2 = -0.375 + 2.640625 Reorder the terms: 2.640625 + -3.25x + x2 = -0.375 + 2.640625 Combine like terms: -0.375 + 2.640625 = 2.265625 2.640625 + -3.25x + x2 = 2.265625 Factor a perfect square on the left side: (x + -1.625)(x + -1.625) = 2.265625 Calculate the square root of the right side: 1.505199322 Break this problem into two subproblems by setting (x + -1.625) equal to 1.505199322 and -1.505199322.Subproblem 1
x + -1.625 = 1.505199322 Simplifying x + -1.625 = 1.505199322 Reorder the terms: -1.625 + x = 1.505199322 Solving -1.625 + x = 1.505199322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + x = 1.505199322 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + x = 1.505199322 + 1.625 x = 1.505199322 + 1.625 Combine like terms: 1.505199322 + 1.625 = 3.130199322 x = 3.130199322 Simplifying x = 3.130199322Subproblem 2
x + -1.625 = -1.505199322 Simplifying x + -1.625 = -1.505199322 Reorder the terms: -1.625 + x = -1.505199322 Solving -1.625 + x = -1.505199322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + x = -1.505199322 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + x = -1.505199322 + 1.625 x = -1.505199322 + 1.625 Combine like terms: -1.505199322 + 1.625 = 0.119800678 x = 0.119800678 Simplifying x = 0.119800678Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.130199322, 0.119800678}Solution
x = {3.130199322, 0.119800678}
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