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Simplifying (2x2) + -1(x4) = 0.5 Solving (2x2) + -1x4 = 0.5 Solving for variable 'x'. Reorder the terms: -0.5 + (2x2) + -1x4 = 0.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 -0.5 + (2x2) + -1x4 = 0.0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 0.5 + (-2x2) + x4 = 0.0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + (-2x2) + -0.5 + x4 = 0.0 + -0.5 Reorder the terms: 0.5 + -0.5 + (-2x2) + x4 = 0.0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + (-2x2) + x4 = 0.0 + -0.5 (-2x2) + x4 = 0.0 + -0.5 Combine like terms: 0.0 + -0.5 = -0.5 (-2x2) + x4 = -0.5 The x term is (-2x2). Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. (-2x2) + 1 + x4 = -0.5 + 1 Reorder the terms: 1 + (-2x2) + x4 = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 1 + (-2x2) + x4 = 0.5 Factor a perfect square on the left side: (x2 + -1)(x2 + -1) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (x2 + -1) equal to 0.707106781 and -0.707106781.Subproblem 1
x2 + -1 = 0.707106781 Simplifying x2 + -1 = 0.707106781 Reorder the terms: -1 + x2 = 0.707106781 Solving -1 + x2 = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x2 = 0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + x2 = 0.707106781 + 1 x2 = 0.707106781 + 1 Combine like terms: 0.707106781 + 1 = 1.707106781 x2 = 1.707106781 Simplifying x2 = 1.707106781 Take the square root of each side: x = {-1.306562965, 1.306562965}Subproblem 2
x2 + -1 = -0.707106781 Simplifying x2 + -1 = -0.707106781 Reorder the terms: -1 + x2 = -0.707106781 Solving -1 + x2 = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x2 = -0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + x2 = -0.707106781 + 1 x2 = -0.707106781 + 1 Combine like terms: -0.707106781 + 1 = 0.292893219 x2 = 0.292893219 Simplifying x2 = 0.292893219 Take the square root of each side: x = {-0.5411961, 0.5411961}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.306562965, 1.306562965, -0.5411961, 0.5411961}
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