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Simplifying 6x2 + 73x + -819 = 1000 Reorder the terms: -819 + 73x + 6x2 = 1000 Solving -819 + 73x + 6x2 = 1000 Solving for variable 'x'. Reorder the terms: -819 + -1000 + 73x + 6x2 = 1000 + -1000 Combine like terms: -819 + -1000 = -1819 -1819 + 73x + 6x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1819 + 73x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -303.1666667 + 12.16666667x + x2 = 0 Move the constant term to the right: Add '303.1666667' to each side of the equation. -303.1666667 + 12.16666667x + 303.1666667 + x2 = 0 + 303.1666667 Reorder the terms: -303.1666667 + 303.1666667 + 12.16666667x + x2 = 0 + 303.1666667 Combine like terms: -303.1666667 + 303.1666667 = 0.0000000 0.0000000 + 12.16666667x + x2 = 0 + 303.1666667 12.16666667x + x2 = 0 + 303.1666667 Combine like terms: 0 + 303.1666667 = 303.1666667 12.16666667x + x2 = 303.1666667 The x term is 12.16666667x. Take half its coefficient (6.083333335). Square it (37.00694446) and add it to both sides. Add '37.00694446' to each side of the equation. 12.16666667x + 37.00694446 + x2 = 303.1666667 + 37.00694446 Reorder the terms: 37.00694446 + 12.16666667x + x2 = 303.1666667 + 37.00694446 Combine like terms: 303.1666667 + 37.00694446 = 340.17361116 37.00694446 + 12.16666667x + x2 = 340.17361116 Factor a perfect square on the left side: (x + 6.083333335)(x + 6.083333335) = 340.17361116 Calculate the square root of the right side: 18.443796007 Break this problem into two subproblems by setting (x + 6.083333335) equal to 18.443796007 and -18.443796007.Subproblem 1
x + 6.083333335 = 18.443796007 Simplifying x + 6.083333335 = 18.443796007 Reorder the terms: 6.083333335 + x = 18.443796007 Solving 6.083333335 + x = 18.443796007 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.083333335' to each side of the equation. 6.083333335 + -6.083333335 + x = 18.443796007 + -6.083333335 Combine like terms: 6.083333335 + -6.083333335 = 0.000000000 0.000000000 + x = 18.443796007 + -6.083333335 x = 18.443796007 + -6.083333335 Combine like terms: 18.443796007 + -6.083333335 = 12.360462672 x = 12.360462672 Simplifying x = 12.360462672Subproblem 2
x + 6.083333335 = -18.443796007 Simplifying x + 6.083333335 = -18.443796007 Reorder the terms: 6.083333335 + x = -18.443796007 Solving 6.083333335 + x = -18.443796007 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.083333335' to each side of the equation. 6.083333335 + -6.083333335 + x = -18.443796007 + -6.083333335 Combine like terms: 6.083333335 + -6.083333335 = 0.000000000 0.000000000 + x = -18.443796007 + -6.083333335 x = -18.443796007 + -6.083333335 Combine like terms: -18.443796007 + -6.083333335 = -24.527129342 x = -24.527129342 Simplifying x = -24.527129342Solution
The solution to the problem is based on the solutions from the subproblems. x = {12.360462672, -24.527129342}
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