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Simplifying 2x4 + -30x2 + 75 = 0 Reorder the terms: 75 + -30x2 + 2x4 = 0 Solving 75 + -30x2 + 2x4 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 37.5 + -15x2 + x4 = 0 Move the constant term to the right: Add '-37.5' to each side of the equation. 37.5 + -15x2 + -37.5 + x4 = 0 + -37.5 Reorder the terms: 37.5 + -37.5 + -15x2 + x4 = 0 + -37.5 Combine like terms: 37.5 + -37.5 = 0.0 0.0 + -15x2 + x4 = 0 + -37.5 -15x2 + x4 = 0 + -37.5 Combine like terms: 0 + -37.5 = -37.5 -15x2 + x4 = -37.5 The x term is -15x2. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15x2 + 56.25 + x4 = -37.5 + 56.25 Reorder the terms: 56.25 + -15x2 + x4 = -37.5 + 56.25 Combine like terms: -37.5 + 56.25 = 18.75 56.25 + -15x2 + x4 = 18.75 Factor a perfect square on the left side: (x2 + -7.5)(x2 + -7.5) = 18.75 Calculate the square root of the right side: 4.330127019 Break this problem into two subproblems by setting (x2 + -7.5) equal to 4.330127019 and -4.330127019.Subproblem 1
x2 + -7.5 = 4.330127019 Simplifying x2 + -7.5 = 4.330127019 Reorder the terms: -7.5 + x2 = 4.330127019 Solving -7.5 + x2 = 4.330127019 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x2 = 4.330127019 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x2 = 4.330127019 + 7.5 x2 = 4.330127019 + 7.5 Combine like terms: 4.330127019 + 7.5 = 11.830127019 x2 = 11.830127019 Simplifying x2 = 11.830127019 Take the square root of each side: x = {-3.439495169, 3.439495169}Subproblem 2
x2 + -7.5 = -4.330127019 Simplifying x2 + -7.5 = -4.330127019 Reorder the terms: -7.5 + x2 = -4.330127019 Solving -7.5 + x2 = -4.330127019 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x2 = -4.330127019 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x2 = -4.330127019 + 7.5 x2 = -4.330127019 + 7.5 Combine like terms: -4.330127019 + 7.5 = 3.169872981 x2 = 3.169872981 Simplifying x2 = 3.169872981 Take the square root of each side: x = {-1.780413711, 1.780413711}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-3.439495169, 3.439495169, -1.780413711, 1.780413711}
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