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Simplifying 4x2 + 32x + 6 + 3x2 + -7x + 10 = 0 Reorder the terms: 6 + 10 + 32x + -7x + 4x2 + 3x2 = 0 Combine like terms: 6 + 10 = 16 16 + 32x + -7x + 4x2 + 3x2 = 0 Combine like terms: 32x + -7x = 25x 16 + 25x + 4x2 + 3x2 = 0 Combine like terms: 4x2 + 3x2 = 7x2 16 + 25x + 7x2 = 0 Solving 16 + 25x + 7x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 2.285714286 + 3.571428571x + x2 = 0 Move the constant term to the right: Add '-2.285714286' to each side of the equation. 2.285714286 + 3.571428571x + -2.285714286 + x2 = 0 + -2.285714286 Reorder the terms: 2.285714286 + -2.285714286 + 3.571428571x + x2 = 0 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + 3.571428571x + x2 = 0 + -2.285714286 3.571428571x + x2 = 0 + -2.285714286 Combine like terms: 0 + -2.285714286 = -2.285714286 3.571428571x + x2 = -2.285714286 The x term is 3.571428571x. Take half its coefficient (1.785714286). Square it (3.188775511) and add it to both sides. Add '3.188775511' to each side of the equation. 3.571428571x + 3.188775511 + x2 = -2.285714286 + 3.188775511 Reorder the terms: 3.188775511 + 3.571428571x + x2 = -2.285714286 + 3.188775511 Combine like terms: -2.285714286 + 3.188775511 = 0.903061225 3.188775511 + 3.571428571x + x2 = 0.903061225 Factor a perfect square on the left side: (x + 1.785714286)(x + 1.785714286) = 0.903061225 Calculate the square root of the right side: 0.950295336 Break this problem into two subproblems by setting (x + 1.785714286) equal to 0.950295336 and -0.950295336.Subproblem 1
x + 1.785714286 = 0.950295336 Simplifying x + 1.785714286 = 0.950295336 Reorder the terms: 1.785714286 + x = 0.950295336 Solving 1.785714286 + x = 0.950295336 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + x = 0.950295336 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + x = 0.950295336 + -1.785714286 x = 0.950295336 + -1.785714286 Combine like terms: 0.950295336 + -1.785714286 = -0.83541895 x = -0.83541895 Simplifying x = -0.83541895Subproblem 2
x + 1.785714286 = -0.950295336 Simplifying x + 1.785714286 = -0.950295336 Reorder the terms: 1.785714286 + x = -0.950295336 Solving 1.785714286 + x = -0.950295336 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + x = -0.950295336 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + x = -0.950295336 + -1.785714286 x = -0.950295336 + -1.785714286 Combine like terms: -0.950295336 + -1.785714286 = -2.736009622 x = -2.736009622 Simplifying x = -2.736009622Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.83541895, -2.736009622}
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