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Simplifying 16x4 + -28x2 + 8 = 0 Reorder the terms: 8 + -28x2 + 16x4 = 0 Solving 8 + -28x2 + 16x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4'. 4(2 + -7x2 + 4x4) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(2 + -7x2 + 4x4)' equal to zero and attempt to solve: Simplifying 2 + -7x2 + 4x4 = 0 Solving 2 + -7x2 + 4x4 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.5 + -1.75x2 + x4 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -1.75x2 + -0.5 + x4 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -1.75x2 + x4 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -1.75x2 + x4 = 0 + -0.5 -1.75x2 + x4 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -1.75x2 + x4 = -0.5 The x term is -1.75x2. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75x2 + 0.765625 + x4 = -0.5 + 0.765625 Reorder the terms: 0.765625 + -1.75x2 + x4 = -0.5 + 0.765625 Combine like terms: -0.5 + 0.765625 = 0.265625 0.765625 + -1.75x2 + x4 = 0.265625 Factor a perfect square on the left side: (x2 + -0.875)(x2 + -0.875) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (x2 + -0.875) equal to 0.515388203 and -0.515388203.Subproblem 1
x2 + -0.875 = 0.515388203 Simplifying x2 + -0.875 = 0.515388203 Reorder the terms: -0.875 + x2 = 0.515388203 Solving -0.875 + x2 = 0.515388203 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x2 = 0.515388203 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x2 = 0.515388203 + 0.875 x2 = 0.515388203 + 0.875 Combine like terms: 0.515388203 + 0.875 = 1.390388203 x2 = 1.390388203 Simplifying x2 = 1.390388203 Take the square root of each side: x = {-1.179147236, 1.179147236}Subproblem 2
x2 + -0.875 = -0.515388203 Simplifying x2 + -0.875 = -0.515388203 Reorder the terms: -0.875 + x2 = -0.515388203 Solving -0.875 + x2 = -0.515388203 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x2 = -0.515388203 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x2 = -0.515388203 + 0.875 x2 = -0.515388203 + 0.875 Combine like terms: -0.515388203 + 0.875 = 0.359611797 x2 = 0.359611797 Simplifying x2 = 0.359611797 Take the square root of each side: x = {-0.59967641, 0.59967641}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.179147236, 1.179147236, -0.59967641, 0.59967641}Solution
x = {-1.179147236, 1.179147236, -0.59967641, 0.59967641}
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