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Simplifying 9x2 + 31x + 25 = 0 Reorder the terms: 25 + 31x + 9x2 = 0 Solving 25 + 31x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 2.777777778 + 3.444444444x + x2 = 0 Move the constant term to the right: Add '-2.777777778' to each side of the equation. 2.777777778 + 3.444444444x + -2.777777778 + x2 = 0 + -2.777777778 Reorder the terms: 2.777777778 + -2.777777778 + 3.444444444x + x2 = 0 + -2.777777778 Combine like terms: 2.777777778 + -2.777777778 = 0.000000000 0.000000000 + 3.444444444x + x2 = 0 + -2.777777778 3.444444444x + x2 = 0 + -2.777777778 Combine like terms: 0 + -2.777777778 = -2.777777778 3.444444444x + x2 = -2.777777778 The x term is 3.444444444x. Take half its coefficient (1.722222222). Square it (2.966049382) and add it to both sides. Add '2.966049382' to each side of the equation. 3.444444444x + 2.966049382 + x2 = -2.777777778 + 2.966049382 Reorder the terms: 2.966049382 + 3.444444444x + x2 = -2.777777778 + 2.966049382 Combine like terms: -2.777777778 + 2.966049382 = 0.188271604 2.966049382 + 3.444444444x + x2 = 0.188271604 Factor a perfect square on the left side: (x + 1.722222222)(x + 1.722222222) = 0.188271604 Calculate the square root of the right side: 0.433902759 Break this problem into two subproblems by setting (x + 1.722222222) equal to 0.433902759 and -0.433902759.Subproblem 1
x + 1.722222222 = 0.433902759 Simplifying x + 1.722222222 = 0.433902759 Reorder the terms: 1.722222222 + x = 0.433902759 Solving 1.722222222 + x = 0.433902759 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.722222222' to each side of the equation. 1.722222222 + -1.722222222 + x = 0.433902759 + -1.722222222 Combine like terms: 1.722222222 + -1.722222222 = 0.000000000 0.000000000 + x = 0.433902759 + -1.722222222 x = 0.433902759 + -1.722222222 Combine like terms: 0.433902759 + -1.722222222 = -1.288319463 x = -1.288319463 Simplifying x = -1.288319463Subproblem 2
x + 1.722222222 = -0.433902759 Simplifying x + 1.722222222 = -0.433902759 Reorder the terms: 1.722222222 + x = -0.433902759 Solving 1.722222222 + x = -0.433902759 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.722222222' to each side of the equation. 1.722222222 + -1.722222222 + x = -0.433902759 + -1.722222222 Combine like terms: 1.722222222 + -1.722222222 = 0.000000000 0.000000000 + x = -0.433902759 + -1.722222222 x = -0.433902759 + -1.722222222 Combine like terms: -0.433902759 + -1.722222222 = -2.156124981 x = -2.156124981 Simplifying x = -2.156124981Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.288319463, -2.156124981}
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