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Simplifying (7x + -4)(3x + 14) = 120 Reorder the terms: (-4 + 7x)(3x + 14) = 120 Reorder the terms: (-4 + 7x)(14 + 3x) = 120 Multiply (-4 + 7x) * (14 + 3x) (-4(14 + 3x) + 7x * (14 + 3x)) = 120 ((14 * -4 + 3x * -4) + 7x * (14 + 3x)) = 120 ((-56 + -12x) + 7x * (14 + 3x)) = 120 (-56 + -12x + (14 * 7x + 3x * 7x)) = 120 (-56 + -12x + (98x + 21x2)) = 120 Combine like terms: -12x + 98x = 86x (-56 + 86x + 21x2) = 120 Solving -56 + 86x + 21x2 = 120 Solving for variable 'x'. Reorder the terms: -56 + -120 + 86x + 21x2 = 120 + -120 Combine like terms: -56 + -120 = -176 -176 + 86x + 21x2 = 120 + -120 Combine like terms: 120 + -120 = 0 -176 + 86x + 21x2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -8.380952381 + 4.095238095x + x2 = 0 Move the constant term to the right: Add '8.380952381' to each side of the equation. -8.380952381 + 4.095238095x + 8.380952381 + x2 = 0 + 8.380952381 Reorder the terms: -8.380952381 + 8.380952381 + 4.095238095x + x2 = 0 + 8.380952381 Combine like terms: -8.380952381 + 8.380952381 = 0.000000000 0.000000000 + 4.095238095x + x2 = 0 + 8.380952381 4.095238095x + x2 = 0 + 8.380952381 Combine like terms: 0 + 8.380952381 = 8.380952381 4.095238095x + x2 = 8.380952381 The x term is 4.095238095x. Take half its coefficient (2.047619048). Square it (4.192743766) and add it to both sides. Add '4.192743766' to each side of the equation. 4.095238095x + 4.192743766 + x2 = 8.380952381 + 4.192743766 Reorder the terms: 4.192743766 + 4.095238095x + x2 = 8.380952381 + 4.192743766 Combine like terms: 8.380952381 + 4.192743766 = 12.573696147 4.192743766 + 4.095238095x + x2 = 12.573696147 Factor a perfect square on the left side: (x + 2.047619048)(x + 2.047619048) = 12.573696147 Calculate the square root of the right side: 3.545940799 Break this problem into two subproblems by setting (x + 2.047619048) equal to 3.545940799 and -3.545940799.Subproblem 1
x + 2.047619048 = 3.545940799 Simplifying x + 2.047619048 = 3.545940799 Reorder the terms: 2.047619048 + x = 3.545940799 Solving 2.047619048 + x = 3.545940799 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.047619048' to each side of the equation. 2.047619048 + -2.047619048 + x = 3.545940799 + -2.047619048 Combine like terms: 2.047619048 + -2.047619048 = 0.000000000 0.000000000 + x = 3.545940799 + -2.047619048 x = 3.545940799 + -2.047619048 Combine like terms: 3.545940799 + -2.047619048 = 1.498321751 x = 1.498321751 Simplifying x = 1.498321751Subproblem 2
x + 2.047619048 = -3.545940799 Simplifying x + 2.047619048 = -3.545940799 Reorder the terms: 2.047619048 + x = -3.545940799 Solving 2.047619048 + x = -3.545940799 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.047619048' to each side of the equation. 2.047619048 + -2.047619048 + x = -3.545940799 + -2.047619048 Combine like terms: 2.047619048 + -2.047619048 = 0.000000000 0.000000000 + x = -3.545940799 + -2.047619048 x = -3.545940799 + -2.047619048 Combine like terms: -3.545940799 + -2.047619048 = -5.593559847 x = -5.593559847 Simplifying x = -5.593559847Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.498321751, -5.593559847}
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