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Simplifying 196x2 + -40x + -108 = 0 Reorder the terms: -108 + -40x + 196x2 = 0 Solving -108 + -40x + 196x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4'. 4(-27 + -10x + 49x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-27 + -10x + 49x2)' equal to zero and attempt to solve: Simplifying -27 + -10x + 49x2 = 0 Solving -27 + -10x + 49x2 = 0 Begin completing the square. Divide all terms by 49 the coefficient of the squared term: Divide each side by '49'. -0.5510204082 + -0.2040816327x + x2 = 0 Move the constant term to the right: Add '0.5510204082' to each side of the equation. -0.5510204082 + -0.2040816327x + 0.5510204082 + x2 = 0 + 0.5510204082 Reorder the terms: -0.5510204082 + 0.5510204082 + -0.2040816327x + x2 = 0 + 0.5510204082 Combine like terms: -0.5510204082 + 0.5510204082 = 0.0000000000 0.0000000000 + -0.2040816327x + x2 = 0 + 0.5510204082 -0.2040816327x + x2 = 0 + 0.5510204082 Combine like terms: 0 + 0.5510204082 = 0.5510204082 -0.2040816327x + x2 = 0.5510204082 The x term is -0.2040816327x. Take half its coefficient (-0.1020408164). Square it (0.01041232821) and add it to both sides. Add '0.01041232821' to each side of the equation. -0.2040816327x + 0.01041232821 + x2 = 0.5510204082 + 0.01041232821 Reorder the terms: 0.01041232821 + -0.2040816327x + x2 = 0.5510204082 + 0.01041232821 Combine like terms: 0.5510204082 + 0.01041232821 = 0.56143273641 0.01041232821 + -0.2040816327x + x2 = 0.56143273641 Factor a perfect square on the left side: (x + -0.1020408164)(x + -0.1020408164) = 0.56143273641 Calculate the square root of the right side: 0.749288153 Break this problem into two subproblems by setting (x + -0.1020408164) equal to 0.749288153 and -0.749288153.Subproblem 1
x + -0.1020408164 = 0.749288153 Simplifying x + -0.1020408164 = 0.749288153 Reorder the terms: -0.1020408164 + x = 0.749288153 Solving -0.1020408164 + x = 0.749288153 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1020408164' to each side of the equation. -0.1020408164 + 0.1020408164 + x = 0.749288153 + 0.1020408164 Combine like terms: -0.1020408164 + 0.1020408164 = 0.0000000000 0.0000000000 + x = 0.749288153 + 0.1020408164 x = 0.749288153 + 0.1020408164 Combine like terms: 0.749288153 + 0.1020408164 = 0.8513289694 x = 0.8513289694 Simplifying x = 0.8513289694Subproblem 2
x + -0.1020408164 = -0.749288153 Simplifying x + -0.1020408164 = -0.749288153 Reorder the terms: -0.1020408164 + x = -0.749288153 Solving -0.1020408164 + x = -0.749288153 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1020408164' to each side of the equation. -0.1020408164 + 0.1020408164 + x = -0.749288153 + 0.1020408164 Combine like terms: -0.1020408164 + 0.1020408164 = 0.0000000000 0.0000000000 + x = -0.749288153 + 0.1020408164 x = -0.749288153 + 0.1020408164 Combine like terms: -0.749288153 + 0.1020408164 = -0.6472473366 x = -0.6472473366 Simplifying x = -0.6472473366Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.8513289694, -0.6472473366}Solution
x = {0.8513289694, -0.6472473366}
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