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Simplifying 4x4 + 2x2 + -1 = 7 Reorder the terms: -1 + 2x2 + 4x4 = 7 Solving -1 + 2x2 + 4x4 = 7 Solving for variable 'x'. Reorder the terms: -1 + -7 + 2x2 + 4x4 = 7 + -7 Combine like terms: -1 + -7 = -8 -8 + 2x2 + 4x4 = 7 + -7 Combine like terms: 7 + -7 = 0 -8 + 2x2 + 4x4 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + x2 + 2x4) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + x2 + 2x4)' equal to zero and attempt to solve: Simplifying -4 + x2 + 2x4 = 0 Solving -4 + x2 + 2x4 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + 0.5x2 + x4 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.5x2 + 2 + x4 = 0 + 2 Reorder the terms: -2 + 2 + 0.5x2 + x4 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.5x2 + x4 = 0 + 2 0.5x2 + x4 = 0 + 2 Combine like terms: 0 + 2 = 2 0.5x2 + x4 = 2 The x term is x2. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5x2 + 0.25 + x4 = 2 + 0.25 Reorder the terms: 0.25 + 0.5x2 + x4 = 2 + 0.25 Combine like terms: 2 + 0.25 = 2.25 0.25 + 0.5x2 + x4 = 2.25 Factor a perfect square on the left side: (x2 + 0.5)(x2 + 0.5) = 2.25 Calculate the square root of the right side: 1.5 Break this problem into two subproblems by setting (x2 + 0.5) equal to 1.5 and -1.5.Subproblem 1
x2 + 0.5 = 1.5 Simplifying x2 + 0.5 = 1.5 Reorder the terms: 0.5 + x2 = 1.5 Solving 0.5 + x2 = 1.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x2 = 1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x2 = 1.5 + -0.5 x2 = 1.5 + -0.5 Combine like terms: 1.5 + -0.5 = 1 x2 = 1 Simplifying x2 = 1 Take the square root of each side: x = {-1, 1}Subproblem 2
x2 + 0.5 = -1.5 Simplifying x2 + 0.5 = -1.5 Reorder the terms: 0.5 + x2 = -1.5 Solving 0.5 + x2 = -1.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x2 = -1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x2 = -1.5 + -0.5 x2 = -1.5 + -0.5 Combine like terms: -1.5 + -0.5 = -2 x2 = -2 Simplifying x2 = -2 Reorder the terms: 2 + x2 = -2 + 2 Combine like terms: -2 + 2 = 0 2 + x2 = 0 The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.
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