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Simplifying 7x2 + 37x + -42 = 1 Reorder the terms: -42 + 37x + 7x2 = 1 Solving -42 + 37x + 7x2 = 1 Solving for variable 'x'. Reorder the terms: -42 + -1 + 37x + 7x2 = 1 + -1 Combine like terms: -42 + -1 = -43 -43 + 37x + 7x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -43 + 37x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -6.142857143 + 5.285714286x + x2 = 0 Move the constant term to the right: Add '6.142857143' to each side of the equation. -6.142857143 + 5.285714286x + 6.142857143 + x2 = 0 + 6.142857143 Reorder the terms: -6.142857143 + 6.142857143 + 5.285714286x + x2 = 0 + 6.142857143 Combine like terms: -6.142857143 + 6.142857143 = 0.000000000 0.000000000 + 5.285714286x + x2 = 0 + 6.142857143 5.285714286x + x2 = 0 + 6.142857143 Combine like terms: 0 + 6.142857143 = 6.142857143 5.285714286x + x2 = 6.142857143 The x term is 5.285714286x. Take half its coefficient (2.642857143). Square it (6.984693878) and add it to both sides. Add '6.984693878' to each side of the equation. 5.285714286x + 6.984693878 + x2 = 6.142857143 + 6.984693878 Reorder the terms: 6.984693878 + 5.285714286x + x2 = 6.142857143 + 6.984693878 Combine like terms: 6.142857143 + 6.984693878 = 13.127551021 6.984693878 + 5.285714286x + x2 = 13.127551021 Factor a perfect square on the left side: (x + 2.642857143)(x + 2.642857143) = 13.127551021 Calculate the square root of the right side: 3.623196244 Break this problem into two subproblems by setting (x + 2.642857143) equal to 3.623196244 and -3.623196244.Subproblem 1
x + 2.642857143 = 3.623196244 Simplifying x + 2.642857143 = 3.623196244 Reorder the terms: 2.642857143 + x = 3.623196244 Solving 2.642857143 + x = 3.623196244 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.642857143' to each side of the equation. 2.642857143 + -2.642857143 + x = 3.623196244 + -2.642857143 Combine like terms: 2.642857143 + -2.642857143 = 0.000000000 0.000000000 + x = 3.623196244 + -2.642857143 x = 3.623196244 + -2.642857143 Combine like terms: 3.623196244 + -2.642857143 = 0.980339101 x = 0.980339101 Simplifying x = 0.980339101Subproblem 2
x + 2.642857143 = -3.623196244 Simplifying x + 2.642857143 = -3.623196244 Reorder the terms: 2.642857143 + x = -3.623196244 Solving 2.642857143 + x = -3.623196244 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.642857143' to each side of the equation. 2.642857143 + -2.642857143 + x = -3.623196244 + -2.642857143 Combine like terms: 2.642857143 + -2.642857143 = 0.000000000 0.000000000 + x = -3.623196244 + -2.642857143 x = -3.623196244 + -2.642857143 Combine like terms: -3.623196244 + -2.642857143 = -6.266053387 x = -6.266053387 Simplifying x = -6.266053387Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.980339101, -6.266053387}
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