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Simplifying 8 = -16x2 + 22x + 6 Reorder the terms: 8 = 6 + 22x + -16x2 Solving 8 = 6 + 22x + -16x2 Solving for variable 'x'. Combine like terms: 8 + -6 = 2 2 + -22x + 16x2 = 6 + 22x + -16x2 + -6 + -22x + 16x2 Reorder the terms: 2 + -22x + 16x2 = 6 + -6 + 22x + -22x + -16x2 + 16x2 Combine like terms: 6 + -6 = 0 2 + -22x + 16x2 = 0 + 22x + -22x + -16x2 + 16x2 2 + -22x + 16x2 = 22x + -22x + -16x2 + 16x2 Combine like terms: 22x + -22x = 0 2 + -22x + 16x2 = 0 + -16x2 + 16x2 2 + -22x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 2 + -22x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(1 + -11x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + -11x + 8x2)' equal to zero and attempt to solve: Simplifying 1 + -11x + 8x2 = 0 Solving 1 + -11x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.125 + -1.375x + x2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + -1.375x + -0.125 + x2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + -1.375x + x2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + -1.375x + x2 = 0 + -0.125 -1.375x + x2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 -1.375x + x2 = -0.125 The x term is -1.375x. Take half its coefficient (-0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. -1.375x + 0.47265625 + x2 = -0.125 + 0.47265625 Reorder the terms: 0.47265625 + -1.375x + x2 = -0.125 + 0.47265625 Combine like terms: -0.125 + 0.47265625 = 0.34765625 0.47265625 + -1.375x + x2 = 0.34765625 Factor a perfect square on the left side: (x + -0.6875)(x + -0.6875) = 0.34765625 Calculate the square root of the right side: 0.589623821 Break this problem into two subproblems by setting (x + -0.6875) equal to 0.589623821 and -0.589623821.Subproblem 1
x + -0.6875 = 0.589623821 Simplifying x + -0.6875 = 0.589623821 Reorder the terms: -0.6875 + x = 0.589623821 Solving -0.6875 + x = 0.589623821 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + x = 0.589623821 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + x = 0.589623821 + 0.6875 x = 0.589623821 + 0.6875 Combine like terms: 0.589623821 + 0.6875 = 1.277123821 x = 1.277123821 Simplifying x = 1.277123821Subproblem 2
x + -0.6875 = -0.589623821 Simplifying x + -0.6875 = -0.589623821 Reorder the terms: -0.6875 + x = -0.589623821 Solving -0.6875 + x = -0.589623821 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + x = -0.589623821 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + x = -0.589623821 + 0.6875 x = -0.589623821 + 0.6875 Combine like terms: -0.589623821 + 0.6875 = 0.097876179 x = 0.097876179 Simplifying x = 0.097876179Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.277123821, 0.097876179}Solution
x = {1.277123821, 0.097876179}
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