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Simplifying 3x4 + -21x2 + 1 = 0 Reorder the terms: 1 + -21x2 + 3x4 = 0 Solving 1 + -21x2 + 3x4 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.3333333333 + -7x2 + x4 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -7x2 + -0.3333333333 + x4 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -7x2 + x4 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -7x2 + x4 = 0 + -0.3333333333 -7x2 + x4 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -7x2 + x4 = -0.3333333333 The x term is -7x2. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7x2 + 12.25 + x4 = -0.3333333333 + 12.25 Reorder the terms: 12.25 + -7x2 + x4 = -0.3333333333 + 12.25 Combine like terms: -0.3333333333 + 12.25 = 11.9166666667 12.25 + -7x2 + x4 = 11.9166666667 Factor a perfect square on the left side: (x2 + -3.5)(x2 + -3.5) = 11.9166666667 Calculate the square root of the right side: 3.45205253 Break this problem into two subproblems by setting (x2 + -3.5) equal to 3.45205253 and -3.45205253.Subproblem 1
x2 + -3.5 = 3.45205253 Simplifying x2 + -3.5 = 3.45205253 Reorder the terms: -3.5 + x2 = 3.45205253 Solving -3.5 + x2 = 3.45205253 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x2 = 3.45205253 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x2 = 3.45205253 + 3.5 x2 = 3.45205253 + 3.5 Combine like terms: 3.45205253 + 3.5 = 6.95205253 x2 = 6.95205253 Simplifying x2 = 6.95205253 Take the square root of each side: x = {-2.636674521, 2.636674521}Subproblem 2
x2 + -3.5 = -3.45205253 Simplifying x2 + -3.5 = -3.45205253 Reorder the terms: -3.5 + x2 = -3.45205253 Solving -3.5 + x2 = -3.45205253 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x2 = -3.45205253 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x2 = -3.45205253 + 3.5 x2 = -3.45205253 + 3.5 Combine like terms: -3.45205253 + 3.5 = 0.04794747 x2 = 0.04794747 Simplifying x2 = 0.04794747 Take the square root of each side: x = {-0.218969107, 0.218969107}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.636674521, 2.636674521, -0.218969107, 0.218969107}
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