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Simplifying 3x2 + 17x + 3 = (x + 1)(x + 13) Reorder the terms: 3 + 17x + 3x2 = (x + 1)(x + 13) Reorder the terms: 3 + 17x + 3x2 = (1 + x)(x + 13) Reorder the terms: 3 + 17x + 3x2 = (1 + x)(13 + x) Multiply (1 + x) * (13 + x) 3 + 17x + 3x2 = (1(13 + x) + x(13 + x)) 3 + 17x + 3x2 = ((13 * 1 + x * 1) + x(13 + x)) 3 + 17x + 3x2 = ((13 + 1x) + x(13 + x)) 3 + 17x + 3x2 = (13 + 1x + (13 * x + x * x)) 3 + 17x + 3x2 = (13 + 1x + (13x + x2)) Combine like terms: 1x + 13x = 14x 3 + 17x + 3x2 = (13 + 14x + x2) Solving 3 + 17x + 3x2 = 13 + 14x + x2 Solving for variable 'x'. Reorder the terms: 3 + -13 + 17x + -14x + 3x2 + -1x2 = 13 + 14x + x2 + -13 + -14x + -1x2 Combine like terms: 3 + -13 = -10 -10 + 17x + -14x + 3x2 + -1x2 = 13 + 14x + x2 + -13 + -14x + -1x2 Combine like terms: 17x + -14x = 3x -10 + 3x + 3x2 + -1x2 = 13 + 14x + x2 + -13 + -14x + -1x2 Combine like terms: 3x2 + -1x2 = 2x2 -10 + 3x + 2x2 = 13 + 14x + x2 + -13 + -14x + -1x2 Reorder the terms: -10 + 3x + 2x2 = 13 + -13 + 14x + -14x + x2 + -1x2 Combine like terms: 13 + -13 = 0 -10 + 3x + 2x2 = 0 + 14x + -14x + x2 + -1x2 -10 + 3x + 2x2 = 14x + -14x + x2 + -1x2 Combine like terms: 14x + -14x = 0 -10 + 3x + 2x2 = 0 + x2 + -1x2 -10 + 3x + 2x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -10 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -5 + 1.5x + x2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 1.5x + 5 + x2 = 0 + 5 Reorder the terms: -5 + 5 + 1.5x + x2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 1.5x + x2 = 0 + 5 1.5x + x2 = 0 + 5 Combine like terms: 0 + 5 = 5 1.5x + x2 = 5 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 5 + 0.5625 Combine like terms: 5 + 0.5625 = 5.5625 0.5625 + 1.5x + x2 = 5.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (x + 0.75) equal to 2.358495283 and -2.358495283.Subproblem 1
x + 0.75 = 2.358495283 Simplifying x + 0.75 = 2.358495283 Reorder the terms: 0.75 + x = 2.358495283 Solving 0.75 + x = 2.358495283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 2.358495283 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 2.358495283 + -0.75 x = 2.358495283 + -0.75 Combine like terms: 2.358495283 + -0.75 = 1.608495283 x = 1.608495283 Simplifying x = 1.608495283Subproblem 2
x + 0.75 = -2.358495283 Simplifying x + 0.75 = -2.358495283 Reorder the terms: 0.75 + x = -2.358495283 Solving 0.75 + x = -2.358495283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -2.358495283 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -2.358495283 + -0.75 x = -2.358495283 + -0.75 Combine like terms: -2.358495283 + -0.75 = -3.108495283 x = -3.108495283 Simplifying x = -3.108495283Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.608495283, -3.108495283}
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