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Simplifying (3x + -24)(4x + 16) + 3x = 14 Reorder the terms: (-24 + 3x)(4x + 16) + 3x = 14 Reorder the terms: (-24 + 3x)(16 + 4x) + 3x = 14 Multiply (-24 + 3x) * (16 + 4x) (-24(16 + 4x) + 3x * (16 + 4x)) + 3x = 14 ((16 * -24 + 4x * -24) + 3x * (16 + 4x)) + 3x = 14 ((-384 + -96x) + 3x * (16 + 4x)) + 3x = 14 (-384 + -96x + (16 * 3x + 4x * 3x)) + 3x = 14 (-384 + -96x + (48x + 12x2)) + 3x = 14 Combine like terms: -96x + 48x = -48x (-384 + -48x + 12x2) + 3x = 14 Reorder the terms: -384 + -48x + 3x + 12x2 = 14 Combine like terms: -48x + 3x = -45x -384 + -45x + 12x2 = 14 Solving -384 + -45x + 12x2 = 14 Solving for variable 'x'. Reorder the terms: -384 + -14 + -45x + 12x2 = 14 + -14 Combine like terms: -384 + -14 = -398 -398 + -45x + 12x2 = 14 + -14 Combine like terms: 14 + -14 = 0 -398 + -45x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -33.16666667 + -3.75x + x2 = 0 Move the constant term to the right: Add '33.16666667' to each side of the equation. -33.16666667 + -3.75x + 33.16666667 + x2 = 0 + 33.16666667 Reorder the terms: -33.16666667 + 33.16666667 + -3.75x + x2 = 0 + 33.16666667 Combine like terms: -33.16666667 + 33.16666667 = 0.00000000 0.00000000 + -3.75x + x2 = 0 + 33.16666667 -3.75x + x2 = 0 + 33.16666667 Combine like terms: 0 + 33.16666667 = 33.16666667 -3.75x + x2 = 33.16666667 The x term is -3.75x. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75x + 3.515625 + x2 = 33.16666667 + 3.515625 Reorder the terms: 3.515625 + -3.75x + x2 = 33.16666667 + 3.515625 Combine like terms: 33.16666667 + 3.515625 = 36.68229167 3.515625 + -3.75x + x2 = 36.68229167 Factor a perfect square on the left side: (x + -1.875)(x + -1.875) = 36.68229167 Calculate the square root of the right side: 6.056590763 Break this problem into two subproblems by setting (x + -1.875) equal to 6.056590763 and -6.056590763.Subproblem 1
x + -1.875 = 6.056590763 Simplifying x + -1.875 = 6.056590763 Reorder the terms: -1.875 + x = 6.056590763 Solving -1.875 + x = 6.056590763 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + x = 6.056590763 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + x = 6.056590763 + 1.875 x = 6.056590763 + 1.875 Combine like terms: 6.056590763 + 1.875 = 7.931590763 x = 7.931590763 Simplifying x = 7.931590763Subproblem 2
x + -1.875 = -6.056590763 Simplifying x + -1.875 = -6.056590763 Reorder the terms: -1.875 + x = -6.056590763 Solving -1.875 + x = -6.056590763 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + x = -6.056590763 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + x = -6.056590763 + 1.875 x = -6.056590763 + 1.875 Combine like terms: -6.056590763 + 1.875 = -4.181590763 x = -4.181590763 Simplifying x = -4.181590763Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.931590763, -4.181590763}
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