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Simplifying (3x + 5)(3x + -20) = 111 Reorder the terms: (5 + 3x)(3x + -20) = 111 Reorder the terms: (5 + 3x)(-20 + 3x) = 111 Multiply (5 + 3x) * (-20 + 3x) (5(-20 + 3x) + 3x * (-20 + 3x)) = 111 ((-20 * 5 + 3x * 5) + 3x * (-20 + 3x)) = 111 ((-100 + 15x) + 3x * (-20 + 3x)) = 111 (-100 + 15x + (-20 * 3x + 3x * 3x)) = 111 (-100 + 15x + (-60x + 9x2)) = 111 Combine like terms: 15x + -60x = -45x (-100 + -45x + 9x2) = 111 Solving -100 + -45x + 9x2 = 111 Solving for variable 'x'. Reorder the terms: -100 + -111 + -45x + 9x2 = 111 + -111 Combine like terms: -100 + -111 = -211 -211 + -45x + 9x2 = 111 + -111 Combine like terms: 111 + -111 = 0 -211 + -45x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -23.44444444 + -5x + x2 = 0 Move the constant term to the right: Add '23.44444444' to each side of the equation. -23.44444444 + -5x + 23.44444444 + x2 = 0 + 23.44444444 Reorder the terms: -23.44444444 + 23.44444444 + -5x + x2 = 0 + 23.44444444 Combine like terms: -23.44444444 + 23.44444444 = 0.00000000 0.00000000 + -5x + x2 = 0 + 23.44444444 -5x + x2 = 0 + 23.44444444 Combine like terms: 0 + 23.44444444 = 23.44444444 -5x + x2 = 23.44444444 The x term is -5x. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5x + 6.25 + x2 = 23.44444444 + 6.25 Reorder the terms: 6.25 + -5x + x2 = 23.44444444 + 6.25 Combine like terms: 23.44444444 + 6.25 = 29.69444444 6.25 + -5x + x2 = 29.69444444 Factor a perfect square on the left side: (x + -2.5)(x + -2.5) = 29.69444444 Calculate the square root of the right side: 5.449260908 Break this problem into two subproblems by setting (x + -2.5) equal to 5.449260908 and -5.449260908.Subproblem 1
x + -2.5 = 5.449260908 Simplifying x + -2.5 = 5.449260908 Reorder the terms: -2.5 + x = 5.449260908 Solving -2.5 + x = 5.449260908 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 5.449260908 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 5.449260908 + 2.5 x = 5.449260908 + 2.5 Combine like terms: 5.449260908 + 2.5 = 7.949260908 x = 7.949260908 Simplifying x = 7.949260908Subproblem 2
x + -2.5 = -5.449260908 Simplifying x + -2.5 = -5.449260908 Reorder the terms: -2.5 + x = -5.449260908 Solving -2.5 + x = -5.449260908 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -5.449260908 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -5.449260908 + 2.5 x = -5.449260908 + 2.5 Combine like terms: -5.449260908 + 2.5 = -2.949260908 x = -2.949260908 Simplifying x = -2.949260908Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.949260908, -2.949260908}
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