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Simplifying 4x2 + 8x + -194 = 0 Reorder the terms: -194 + 8x + 4x2 = 0 Solving -194 + 8x + 4x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-97 + 4x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-97 + 4x + 2x2)' equal to zero and attempt to solve: Simplifying -97 + 4x + 2x2 = 0 Solving -97 + 4x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -48.5 + 2x + x2 = 0 Move the constant term to the right: Add '48.5' to each side of the equation. -48.5 + 2x + 48.5 + x2 = 0 + 48.5 Reorder the terms: -48.5 + 48.5 + 2x + x2 = 0 + 48.5 Combine like terms: -48.5 + 48.5 = 0.0 0.0 + 2x + x2 = 0 + 48.5 2x + x2 = 0 + 48.5 Combine like terms: 0 + 48.5 = 48.5 2x + x2 = 48.5 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 48.5 + 1 Reorder the terms: 1 + 2x + x2 = 48.5 + 1 Combine like terms: 48.5 + 1 = 49.5 1 + 2x + x2 = 49.5 Factor a perfect square on the left side: (x + 1)(x + 1) = 49.5 Calculate the square root of the right side: 7.03562364 Break this problem into two subproblems by setting (x + 1) equal to 7.03562364 and -7.03562364.Subproblem 1
x + 1 = 7.03562364 Simplifying x + 1 = 7.03562364 Reorder the terms: 1 + x = 7.03562364 Solving 1 + x = 7.03562364 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 + x = 7.03562364 + -1 Combine like terms: 1 + -1 = 0 0 + x = 7.03562364 + -1 x = 7.03562364 + -1 Combine like terms: 7.03562364 + -1 = 6.03562364 x = 6.03562364 Simplifying x = 6.03562364Subproblem 2
x + 1 = -7.03562364 Simplifying x + 1 = -7.03562364 Reorder the terms: 1 + x = -7.03562364 Solving 1 + x = -7.03562364 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 + x = -7.03562364 + -1 Combine like terms: 1 + -1 = 0 0 + x = -7.03562364 + -1 x = -7.03562364 + -1 Combine like terms: -7.03562364 + -1 = -8.03562364 x = -8.03562364 Simplifying x = -8.03562364Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.03562364, -8.03562364}Solution
x = {6.03562364, -8.03562364}
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