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Simplifying (3x + 14)(2x + -7) = 100 Reorder the terms: (14 + 3x)(2x + -7) = 100 Reorder the terms: (14 + 3x)(-7 + 2x) = 100 Multiply (14 + 3x) * (-7 + 2x) (14(-7 + 2x) + 3x * (-7 + 2x)) = 100 ((-7 * 14 + 2x * 14) + 3x * (-7 + 2x)) = 100 ((-98 + 28x) + 3x * (-7 + 2x)) = 100 (-98 + 28x + (-7 * 3x + 2x * 3x)) = 100 (-98 + 28x + (-21x + 6x2)) = 100 Combine like terms: 28x + -21x = 7x (-98 + 7x + 6x2) = 100 Solving -98 + 7x + 6x2 = 100 Solving for variable 'x'. Reorder the terms: -98 + -100 + 7x + 6x2 = 100 + -100 Combine like terms: -98 + -100 = -198 -198 + 7x + 6x2 = 100 + -100 Combine like terms: 100 + -100 = 0 -198 + 7x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -33 + 1.166666667x + x2 = 0 Move the constant term to the right: Add '33' to each side of the equation. -33 + 1.166666667x + 33 + x2 = 0 + 33 Reorder the terms: -33 + 33 + 1.166666667x + x2 = 0 + 33 Combine like terms: -33 + 33 = 0 0 + 1.166666667x + x2 = 0 + 33 1.166666667x + x2 = 0 + 33 Combine like terms: 0 + 33 = 33 1.166666667x + x2 = 33 The x term is 1.166666667x. Take half its coefficient (0.5833333335). Square it (0.3402777780) and add it to both sides. Add '0.3402777780' to each side of the equation. 1.166666667x + 0.3402777780 + x2 = 33 + 0.3402777780 Reorder the terms: 0.3402777780 + 1.166666667x + x2 = 33 + 0.3402777780 Combine like terms: 33 + 0.3402777780 = 33.340277778 0.3402777780 + 1.166666667x + x2 = 33.340277778 Factor a perfect square on the left side: (x + 0.5833333335)(x + 0.5833333335) = 33.340277778 Calculate the square root of the right side: 5.774104067 Break this problem into two subproblems by setting (x + 0.5833333335) equal to 5.774104067 and -5.774104067.Subproblem 1
x + 0.5833333335 = 5.774104067 Simplifying x + 0.5833333335 = 5.774104067 Reorder the terms: 0.5833333335 + x = 5.774104067 Solving 0.5833333335 + x = 5.774104067 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + x = 5.774104067 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + x = 5.774104067 + -0.5833333335 x = 5.774104067 + -0.5833333335 Combine like terms: 5.774104067 + -0.5833333335 = 5.1907707335 x = 5.1907707335 Simplifying x = 5.1907707335Subproblem 2
x + 0.5833333335 = -5.774104067 Simplifying x + 0.5833333335 = -5.774104067 Reorder the terms: 0.5833333335 + x = -5.774104067 Solving 0.5833333335 + x = -5.774104067 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + x = -5.774104067 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + x = -5.774104067 + -0.5833333335 x = -5.774104067 + -0.5833333335 Combine like terms: -5.774104067 + -0.5833333335 = -6.3574374005 x = -6.3574374005 Simplifying x = -6.3574374005Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.1907707335, -6.3574374005}
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