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Simplifying 9x4 + -31x2 + 25 = 0 Reorder the terms: 25 + -31x2 + 9x4 = 0 Solving 25 + -31x2 + 9x4 = 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 + -3.444444444x2 + x4 = 0 Move the constant term to the right: Add '-2.777777778' to each side of the equation. 2.777777778 + -3.444444444x2 + -2.777777778 + x4 = 0 + -2.777777778 Reorder the terms: 2.777777778 + -2.777777778 + -3.444444444x2 + x4 = 0 + -2.777777778 Combine like terms: 2.777777778 + -2.777777778 = 0.000000000 0.000000000 + -3.444444444x2 + x4 = 0 + -2.777777778 -3.444444444x2 + x4 = 0 + -2.777777778 Combine like terms: 0 + -2.777777778 = -2.777777778 -3.444444444x2 + x4 = -2.777777778 The x term is -3.444444444x2. 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. -3.444444444x2 + 2.966049382 + x4 = -2.777777778 + 2.966049382 Reorder the terms: 2.966049382 + -3.444444444x2 + x4 = -2.777777778 + 2.966049382 Combine like terms: -2.777777778 + 2.966049382 = 0.188271604 2.966049382 + -3.444444444x2 + x4 = 0.188271604 Factor a perfect square on the left side: (x2 + -1.722222222)(x2 + -1.722222222) = 0.188271604 Calculate the square root of the right side: 0.433902759 Break this problem into two subproblems by setting (x2 + -1.722222222) equal to 0.433902759 and -0.433902759.Subproblem 1
x2 + -1.722222222 = 0.433902759 Simplifying x2 + -1.722222222 = 0.433902759 Reorder the terms: -1.722222222 + x2 = 0.433902759 Solving -1.722222222 + x2 = 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 + x2 = 0.433902759 + 1.722222222 Combine like terms: -1.722222222 + 1.722222222 = 0.000000000 0.000000000 + x2 = 0.433902759 + 1.722222222 x2 = 0.433902759 + 1.722222222 Combine like terms: 0.433902759 + 1.722222222 = 2.156124981 x2 = 2.156124981 Simplifying x2 = 2.156124981 Take the square root of each side: x = {-1.468374946, 1.468374946}Subproblem 2
x2 + -1.722222222 = -0.433902759 Simplifying x2 + -1.722222222 = -0.433902759 Reorder the terms: -1.722222222 + x2 = -0.433902759 Solving -1.722222222 + x2 = -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 + x2 = -0.433902759 + 1.722222222 Combine like terms: -1.722222222 + 1.722222222 = 0.000000000 0.000000000 + x2 = -0.433902759 + 1.722222222 x2 = -0.433902759 + 1.722222222 Combine like terms: -0.433902759 + 1.722222222 = 1.288319463 x2 = 1.288319463 Simplifying x2 = 1.288319463 Take the square root of each side: x = {-1.135041613, 1.135041613}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.468374946, 1.468374946, -1.135041613, 1.135041613}
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