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Simplifying 5(3x2 + 1) + 16(3x + -1) + 3 = 0 Reorder the terms: 5(1 + 3x2) + 16(3x + -1) + 3 = 0 (1 * 5 + 3x2 * 5) + 16(3x + -1) + 3 = 0 (5 + 15x2) + 16(3x + -1) + 3 = 0 Reorder the terms: 5 + 15x2 + 16(-1 + 3x) + 3 = 0 5 + 15x2 + (-1 * 16 + 3x * 16) + 3 = 0 5 + 15x2 + (-16 + 48x) + 3 = 0 Reorder the terms: 5 + -16 + 3 + 48x + 15x2 = 0 Combine like terms: 5 + -16 = -11 -11 + 3 + 48x + 15x2 = 0 Combine like terms: -11 + 3 = -8 -8 + 48x + 15x2 = 0 Solving -8 + 48x + 15x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.5333333333 + 3.2x + x2 = 0 Move the constant term to the right: Add '0.5333333333' to each side of the equation. -0.5333333333 + 3.2x + 0.5333333333 + x2 = 0 + 0.5333333333 Reorder the terms: -0.5333333333 + 0.5333333333 + 3.2x + x2 = 0 + 0.5333333333 Combine like terms: -0.5333333333 + 0.5333333333 = 0.0000000000 0.0000000000 + 3.2x + x2 = 0 + 0.5333333333 3.2x + x2 = 0 + 0.5333333333 Combine like terms: 0 + 0.5333333333 = 0.5333333333 3.2x + x2 = 0.5333333333 The x term is 3.2x. Take half its coefficient (1.6). Square it (2.56) and add it to both sides. Add '2.56' to each side of the equation. 3.2x + 2.56 + x2 = 0.5333333333 + 2.56 Reorder the terms: 2.56 + 3.2x + x2 = 0.5333333333 + 2.56 Combine like terms: 0.5333333333 + 2.56 = 3.0933333333 2.56 + 3.2x + x2 = 3.0933333333 Factor a perfect square on the left side: (x + 1.6)(x + 1.6) = 3.0933333333 Calculate the square root of the right side: 1.758787461 Break this problem into two subproblems by setting (x + 1.6) equal to 1.758787461 and -1.758787461.Subproblem 1
x + 1.6 = 1.758787461 Simplifying x + 1.6 = 1.758787461 Reorder the terms: 1.6 + x = 1.758787461 Solving 1.6 + x = 1.758787461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.6' to each side of the equation. 1.6 + -1.6 + x = 1.758787461 + -1.6 Combine like terms: 1.6 + -1.6 = 0.0 0.0 + x = 1.758787461 + -1.6 x = 1.758787461 + -1.6 Combine like terms: 1.758787461 + -1.6 = 0.158787461 x = 0.158787461 Simplifying x = 0.158787461Subproblem 2
x + 1.6 = -1.758787461 Simplifying x + 1.6 = -1.758787461 Reorder the terms: 1.6 + x = -1.758787461 Solving 1.6 + x = -1.758787461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.6' to each side of the equation. 1.6 + -1.6 + x = -1.758787461 + -1.6 Combine like terms: 1.6 + -1.6 = 0.0 0.0 + x = -1.758787461 + -1.6 x = -1.758787461 + -1.6 Combine like terms: -1.758787461 + -1.6 = -3.358787461 x = -3.358787461 Simplifying x = -3.358787461Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.158787461, -3.358787461}
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