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Simplifying x4 + -16x2 + 65 = 2x2 + -6 Reorder the terms: 65 + -16x2 + x4 = 2x2 + -6 Reorder the terms: 65 + -16x2 + x4 = -6 + 2x2 Solving 65 + -16x2 + x4 = -6 + 2x2 Solving for variable 'x'. Reorder the terms: 65 + 6 + -16x2 + -2x2 + x4 = -6 + 2x2 + 6 + -2x2 Combine like terms: 65 + 6 = 71 71 + -16x2 + -2x2 + x4 = -6 + 2x2 + 6 + -2x2 Combine like terms: -16x2 + -2x2 = -18x2 71 + -18x2 + x4 = -6 + 2x2 + 6 + -2x2 Reorder the terms: 71 + -18x2 + x4 = -6 + 6 + 2x2 + -2x2 Combine like terms: -6 + 6 = 0 71 + -18x2 + x4 = 0 + 2x2 + -2x2 71 + -18x2 + x4 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 71 + -18x2 + x4 = 0 Begin completing the square. Move the constant term to the right: Add '-71' to each side of the equation. 71 + -18x2 + -71 + x4 = 0 + -71 Reorder the terms: 71 + -71 + -18x2 + x4 = 0 + -71 Combine like terms: 71 + -71 = 0 0 + -18x2 + x4 = 0 + -71 -18x2 + x4 = 0 + -71 Combine like terms: 0 + -71 = -71 -18x2 + x4 = -71 The x term is -18x2. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18x2 + 81 + x4 = -71 + 81 Reorder the terms: 81 + -18x2 + x4 = -71 + 81 Combine like terms: -71 + 81 = 10 81 + -18x2 + x4 = 10 Factor a perfect square on the left side: (x2 + -9)(x2 + -9) = 10 Calculate the square root of the right side: 3.16227766 Break this problem into two subproblems by setting (x2 + -9) equal to 3.16227766 and -3.16227766.Subproblem 1
x2 + -9 = 3.16227766 Simplifying x2 + -9 = 3.16227766 Reorder the terms: -9 + x2 = 3.16227766 Solving -9 + x2 = 3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x2 = 3.16227766 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = 3.16227766 + 9 x2 = 3.16227766 + 9 Combine like terms: 3.16227766 + 9 = 12.16227766 x2 = 12.16227766 Simplifying x2 = 12.16227766 Take the square root of each side: x = {-3.487445721, 3.487445721}Subproblem 2
x2 + -9 = -3.16227766 Simplifying x2 + -9 = -3.16227766 Reorder the terms: -9 + x2 = -3.16227766 Solving -9 + x2 = -3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x2 = -3.16227766 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = -3.16227766 + 9 x2 = -3.16227766 + 9 Combine like terms: -3.16227766 + 9 = 5.83772234 x2 = 5.83772234 Simplifying x2 = 5.83772234 Take the square root of each side: x = {-2.416137898, 2.416137898}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-3.487445721, 3.487445721, -2.416137898, 2.416137898}
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