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Simplifying x4 + -8x2 + 0.5 = -15 Reorder the terms: 0.5 + -8x2 + x4 = -15 Solving 0.5 + -8x2 + x4 = -15 Solving for variable 'x'. Reorder the terms: 0.5 + 15 + -8x2 + x4 = -15 + 15 Combine like terms: 0.5 + 15 = 15.5 15.5 + -8x2 + x4 = -15 + 15 Combine like terms: -15 + 15 = 0 15.5 + -8x2 + x4 = 0 Begin completing the square. Move the constant term to the right: Add '-15.5' to each side of the equation. 15.5 + -8x2 + -15.5 + x4 = 0 + -15.5 Reorder the terms: 15.5 + -15.5 + -8x2 + x4 = 0 + -15.5 Combine like terms: 15.5 + -15.5 = 0.0 0.0 + -8x2 + x4 = 0 + -15.5 -8x2 + x4 = 0 + -15.5 Combine like terms: 0 + -15.5 = -15.5 -8x2 + x4 = -15.5 The x term is -8x2. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8x2 + 16 + x4 = -15.5 + 16 Reorder the terms: 16 + -8x2 + x4 = -15.5 + 16 Combine like terms: -15.5 + 16 = 0.5 16 + -8x2 + x4 = 0.5 Factor a perfect square on the left side: (x2 + -4)(x2 + -4) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (x2 + -4) equal to 0.707106781 and -0.707106781.Subproblem 1
x2 + -4 = 0.707106781 Simplifying x2 + -4 = 0.707106781 Reorder the terms: -4 + x2 = 0.707106781 Solving -4 + x2 = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = 0.707106781 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = 0.707106781 + 4 x2 = 0.707106781 + 4 Combine like terms: 0.707106781 + 4 = 4.707106781 x2 = 4.707106781 Simplifying x2 = 4.707106781 Take the square root of each side: x = {-2.169586777, 2.169586777}Subproblem 2
x2 + -4 = -0.707106781 Simplifying x2 + -4 = -0.707106781 Reorder the terms: -4 + x2 = -0.707106781 Solving -4 + x2 = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = -0.707106781 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = -0.707106781 + 4 x2 = -0.707106781 + 4 Combine like terms: -0.707106781 + 4 = 3.292893219 x2 = 3.292893219 Simplifying x2 = 3.292893219 Take the square root of each side: x = {-1.814633081, 1.814633081}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.169586777, 2.169586777, -1.814633081, 1.814633081}
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