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Simplifying 3x2 + 20x + 7 = -4x2 + 4x + 24 Reorder the terms: 7 + 20x + 3x2 = -4x2 + 4x + 24 Reorder the terms: 7 + 20x + 3x2 = 24 + 4x + -4x2 Solving 7 + 20x + 3x2 = 24 + 4x + -4x2 Solving for variable 'x'. Reorder the terms: 7 + -24 + 20x + -4x + 3x2 + 4x2 = 24 + 4x + -4x2 + -24 + -4x + 4x2 Combine like terms: 7 + -24 = -17 -17 + 20x + -4x + 3x2 + 4x2 = 24 + 4x + -4x2 + -24 + -4x + 4x2 Combine like terms: 20x + -4x = 16x -17 + 16x + 3x2 + 4x2 = 24 + 4x + -4x2 + -24 + -4x + 4x2 Combine like terms: 3x2 + 4x2 = 7x2 -17 + 16x + 7x2 = 24 + 4x + -4x2 + -24 + -4x + 4x2 Reorder the terms: -17 + 16x + 7x2 = 24 + -24 + 4x + -4x + -4x2 + 4x2 Combine like terms: 24 + -24 = 0 -17 + 16x + 7x2 = 0 + 4x + -4x + -4x2 + 4x2 -17 + 16x + 7x2 = 4x + -4x + -4x2 + 4x2 Combine like terms: 4x + -4x = 0 -17 + 16x + 7x2 = 0 + -4x2 + 4x2 -17 + 16x + 7x2 = -4x2 + 4x2 Combine like terms: -4x2 + 4x2 = 0 -17 + 16x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.428571429 + 2.285714286x + x2 = 0 Move the constant term to the right: Add '2.428571429' to each side of the equation. -2.428571429 + 2.285714286x + 2.428571429 + x2 = 0 + 2.428571429 Reorder the terms: -2.428571429 + 2.428571429 + 2.285714286x + x2 = 0 + 2.428571429 Combine like terms: -2.428571429 + 2.428571429 = 0.000000000 0.000000000 + 2.285714286x + x2 = 0 + 2.428571429 2.285714286x + x2 = 0 + 2.428571429 Combine like terms: 0 + 2.428571429 = 2.428571429 2.285714286x + x2 = 2.428571429 The x term is 2.285714286x. Take half its coefficient (1.142857143). Square it (1.306122449) and add it to both sides. Add '1.306122449' to each side of the equation. 2.285714286x + 1.306122449 + x2 = 2.428571429 + 1.306122449 Reorder the terms: 1.306122449 + 2.285714286x + x2 = 2.428571429 + 1.306122449 Combine like terms: 2.428571429 + 1.306122449 = 3.734693878 1.306122449 + 2.285714286x + x2 = 3.734693878 Factor a perfect square on the left side: (x + 1.142857143)(x + 1.142857143) = 3.734693878 Calculate the square root of the right side: 1.932535608 Break this problem into two subproblems by setting (x + 1.142857143) equal to 1.932535608 and -1.932535608.Subproblem 1
x + 1.142857143 = 1.932535608 Simplifying x + 1.142857143 = 1.932535608 Reorder the terms: 1.142857143 + x = 1.932535608 Solving 1.142857143 + x = 1.932535608 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.142857143' to each side of the equation. 1.142857143 + -1.142857143 + x = 1.932535608 + -1.142857143 Combine like terms: 1.142857143 + -1.142857143 = 0.000000000 0.000000000 + x = 1.932535608 + -1.142857143 x = 1.932535608 + -1.142857143 Combine like terms: 1.932535608 + -1.142857143 = 0.789678465 x = 0.789678465 Simplifying x = 0.789678465Subproblem 2
x + 1.142857143 = -1.932535608 Simplifying x + 1.142857143 = -1.932535608 Reorder the terms: 1.142857143 + x = -1.932535608 Solving 1.142857143 + x = -1.932535608 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.142857143' to each side of the equation. 1.142857143 + -1.142857143 + x = -1.932535608 + -1.142857143 Combine like terms: 1.142857143 + -1.142857143 = 0.000000000 0.000000000 + x = -1.932535608 + -1.142857143 x = -1.932535608 + -1.142857143 Combine like terms: -1.932535608 + -1.142857143 = -3.075392751 x = -3.075392751 Simplifying x = -3.075392751Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.789678465, -3.075392751}
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