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Simplifying 4x4 + -31x2 + 56 = 0 Reorder the terms: 56 + -31x2 + 4x4 = 0 Solving 56 + -31x2 + 4x4 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 14 + -7.75x2 + x4 = 0 Move the constant term to the right: Add '-14' to each side of the equation. 14 + -7.75x2 + -14 + x4 = 0 + -14 Reorder the terms: 14 + -14 + -7.75x2 + x4 = 0 + -14 Combine like terms: 14 + -14 = 0 0 + -7.75x2 + x4 = 0 + -14 -7.75x2 + x4 = 0 + -14 Combine like terms: 0 + -14 = -14 -7.75x2 + x4 = -14 The x term is -7.75x2. Take half its coefficient (-3.875). Square it (15.015625) and add it to both sides. Add '15.015625' to each side of the equation. -7.75x2 + 15.015625 + x4 = -14 + 15.015625 Reorder the terms: 15.015625 + -7.75x2 + x4 = -14 + 15.015625 Combine like terms: -14 + 15.015625 = 1.015625 15.015625 + -7.75x2 + x4 = 1.015625 Factor a perfect square on the left side: (x2 + -3.875)(x2 + -3.875) = 1.015625 Calculate the square root of the right side: 1.007782219 Break this problem into two subproblems by setting (x2 + -3.875) equal to 1.007782219 and -1.007782219.Subproblem 1
x2 + -3.875 = 1.007782219 Simplifying x2 + -3.875 = 1.007782219 Reorder the terms: -3.875 + x2 = 1.007782219 Solving -3.875 + x2 = 1.007782219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + x2 = 1.007782219 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + x2 = 1.007782219 + 3.875 x2 = 1.007782219 + 3.875 Combine like terms: 1.007782219 + 3.875 = 4.882782219 x2 = 4.882782219 Simplifying x2 = 4.882782219 Take the square root of each side: x = {-2.209701839, 2.209701839}Subproblem 2
x2 + -3.875 = -1.007782219 Simplifying x2 + -3.875 = -1.007782219 Reorder the terms: -3.875 + x2 = -1.007782219 Solving -3.875 + x2 = -1.007782219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + x2 = -1.007782219 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + x2 = -1.007782219 + 3.875 x2 = -1.007782219 + 3.875 Combine like terms: -1.007782219 + 3.875 = 2.867217781 x2 = 2.867217781 Simplifying x2 = 2.867217781 Take the square root of each side: x = {-1.69328609, 1.69328609}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.209701839, 2.209701839, -1.69328609, 1.69328609}
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