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Simplifying 5x(1 + -2x) + -1x(3x + -32) + 65 = 0 (1 * 5x + -2x * 5x) + -1x(3x + -32) + 65 = 0 (5x + -10x2) + -1x(3x + -32) + 65 = 0 Reorder the terms: 5x + -10x2 + -1x(-32 + 3x) + 65 = 0 5x + -10x2 + (-32 * -1x + 3x * -1x) + 65 = 0 5x + -10x2 + (32x + -3x2) + 65 = 0 Reorder the terms: 65 + 5x + 32x + -10x2 + -3x2 = 0 Combine like terms: 5x + 32x = 37x 65 + 37x + -10x2 + -3x2 = 0 Combine like terms: -10x2 + -3x2 = -13x2 65 + 37x + -13x2 = 0 Solving 65 + 37x + -13x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -13 the coefficient of the squared term: Divide each side by '-13'. -5 + -2.846153846x + x2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -2.846153846x + 5 + x2 = 0 + 5 Reorder the terms: -5 + 5 + -2.846153846x + x2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2.846153846x + x2 = 0 + 5 -2.846153846x + x2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2.846153846x + x2 = 5 The x term is -2.846153846x. Take half its coefficient (-1.423076923). Square it (2.025147929) and add it to both sides. Add '2.025147929' to each side of the equation. -2.846153846x + 2.025147929 + x2 = 5 + 2.025147929 Reorder the terms: 2.025147929 + -2.846153846x + x2 = 5 + 2.025147929 Combine like terms: 5 + 2.025147929 = 7.025147929 2.025147929 + -2.846153846x + x2 = 7.025147929 Factor a perfect square on the left side: (x + -1.423076923)(x + -1.423076923) = 7.025147929 Calculate the square root of the right side: 2.650499562 Break this problem into two subproblems by setting (x + -1.423076923) equal to 2.650499562 and -2.650499562.Subproblem 1
x + -1.423076923 = 2.650499562 Simplifying x + -1.423076923 = 2.650499562 Reorder the terms: -1.423076923 + x = 2.650499562 Solving -1.423076923 + x = 2.650499562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.423076923' to each side of the equation. -1.423076923 + 1.423076923 + x = 2.650499562 + 1.423076923 Combine like terms: -1.423076923 + 1.423076923 = 0.000000000 0.000000000 + x = 2.650499562 + 1.423076923 x = 2.650499562 + 1.423076923 Combine like terms: 2.650499562 + 1.423076923 = 4.073576485 x = 4.073576485 Simplifying x = 4.073576485Subproblem 2
x + -1.423076923 = -2.650499562 Simplifying x + -1.423076923 = -2.650499562 Reorder the terms: -1.423076923 + x = -2.650499562 Solving -1.423076923 + x = -2.650499562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.423076923' to each side of the equation. -1.423076923 + 1.423076923 + x = -2.650499562 + 1.423076923 Combine like terms: -1.423076923 + 1.423076923 = 0.000000000 0.000000000 + x = -2.650499562 + 1.423076923 x = -2.650499562 + 1.423076923 Combine like terms: -2.650499562 + 1.423076923 = -1.227422639 x = -1.227422639 Simplifying x = -1.227422639Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.073576485, -1.227422639}
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