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Simplifying (2x + -3)(3x + 5) = 14 Reorder the terms: (-3 + 2x)(3x + 5) = 14 Reorder the terms: (-3 + 2x)(5 + 3x) = 14 Multiply (-3 + 2x) * (5 + 3x) (-3(5 + 3x) + 2x * (5 + 3x)) = 14 ((5 * -3 + 3x * -3) + 2x * (5 + 3x)) = 14 ((-15 + -9x) + 2x * (5 + 3x)) = 14 (-15 + -9x + (5 * 2x + 3x * 2x)) = 14 (-15 + -9x + (10x + 6x2)) = 14 Combine like terms: -9x + 10x = 1x (-15 + 1x + 6x2) = 14 Solving -15 + 1x + 6x2 = 14 Solving for variable 'x'. Reorder the terms: -15 + -14 + 1x + 6x2 = 14 + -14 Combine like terms: -15 + -14 = -29 -29 + 1x + 6x2 = 14 + -14 Combine like terms: 14 + -14 = 0 -29 + 1x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -4.833333333 + 0.1666666667x + x2 = 0 Move the constant term to the right: Add '4.833333333' to each side of the equation. -4.833333333 + 0.1666666667x + 4.833333333 + x2 = 0 + 4.833333333 Reorder the terms: -4.833333333 + 4.833333333 + 0.1666666667x + x2 = 0 + 4.833333333 Combine like terms: -4.833333333 + 4.833333333 = 0.000000000 0.000000000 + 0.1666666667x + x2 = 0 + 4.833333333 0.1666666667x + x2 = 0 + 4.833333333 Combine like terms: 0 + 4.833333333 = 4.833333333 0.1666666667x + x2 = 4.833333333 The x term is 0.1666666667x. Take half its coefficient (0.08333333335). Square it (0.006944444447) and add it to both sides. Add '0.006944444447' to each side of the equation. 0.1666666667x + 0.006944444447 + x2 = 4.833333333 + 0.006944444447 Reorder the terms: 0.006944444447 + 0.1666666667x + x2 = 4.833333333 + 0.006944444447 Combine like terms: 4.833333333 + 0.006944444447 = 4.840277777447 0.006944444447 + 0.1666666667x + x2 = 4.840277777447 Factor a perfect square on the left side: (x + 0.08333333335)(x + 0.08333333335) = 4.840277777447 Calculate the square root of the right side: 2.20006313 Break this problem into two subproblems by setting (x + 0.08333333335) equal to 2.20006313 and -2.20006313.Subproblem 1
x + 0.08333333335 = 2.20006313 Simplifying x + 0.08333333335 = 2.20006313 Reorder the terms: 0.08333333335 + x = 2.20006313 Solving 0.08333333335 + x = 2.20006313 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = 2.20006313 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = 2.20006313 + -0.08333333335 x = 2.20006313 + -0.08333333335 Combine like terms: 2.20006313 + -0.08333333335 = 2.11672979665 x = 2.11672979665 Simplifying x = 2.11672979665Subproblem 2
x + 0.08333333335 = -2.20006313 Simplifying x + 0.08333333335 = -2.20006313 Reorder the terms: 0.08333333335 + x = -2.20006313 Solving 0.08333333335 + x = -2.20006313 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = -2.20006313 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = -2.20006313 + -0.08333333335 x = -2.20006313 + -0.08333333335 Combine like terms: -2.20006313 + -0.08333333335 = -2.28339646335 x = -2.28339646335 Simplifying x = -2.28339646335Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.11672979665, -2.28339646335}
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