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Simplifying 0.25x4 + -3.25x2 + 9 = 0 Reorder the terms: 9 + -3.25x2 + 0.25x4 = 0 Solving 9 + -3.25x2 + 0.25x4 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.25 the coefficient of the squared term: Divide each side by '0.25'. 36 + -13x2 + x4 = 0 Move the constant term to the right: Add '-36' to each side of the equation. 36 + -13x2 + -36 + x4 = 0 + -36 Reorder the terms: 36 + -36 + -13x2 + x4 = 0 + -36 Combine like terms: 36 + -36 = 0 0 + -13x2 + x4 = 0 + -36 -13x2 + x4 = 0 + -36 Combine like terms: 0 + -36 = -36 -13x2 + x4 = -36 The x term is -13x2. Take half its coefficient (-6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. -13x2 + 42.25 + x4 = -36 + 42.25 Reorder the terms: 42.25 + -13x2 + x4 = -36 + 42.25 Combine like terms: -36 + 42.25 = 6.25 42.25 + -13x2 + x4 = 6.25 Factor a perfect square on the left side: (x2 + -6.5)(x2 + -6.5) = 6.25 Calculate the square root of the right side: 2.5 Break this problem into two subproblems by setting (x2 + -6.5) equal to 2.5 and -2.5.Subproblem 1
x2 + -6.5 = 2.5 Simplifying x2 + -6.5 = 2.5 Reorder the terms: -6.5 + x2 = 2.5 Solving -6.5 + x2 = 2.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x2 = 2.5 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x2 = 2.5 + 6.5 x2 = 2.5 + 6.5 Combine like terms: 2.5 + 6.5 = 9 x2 = 9 Simplifying x2 = 9 Take the square root of each side: x = {-3, 3}Subproblem 2
x2 + -6.5 = -2.5 Simplifying x2 + -6.5 = -2.5 Reorder the terms: -6.5 + x2 = -2.5 Solving -6.5 + x2 = -2.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x2 = -2.5 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x2 = -2.5 + 6.5 x2 = -2.5 + 6.5 Combine like terms: -2.5 + 6.5 = 4 x2 = 4 Simplifying x2 = 4 Take the square root of each side: x = {-2, 2}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-3, 3, -2, 2}
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