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Simplifying 64x2 + 144x + 78 = 0 Reorder the terms: 78 + 144x + 64x2 = 0 Solving 78 + 144x + 64x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(39 + 72x + 32x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(39 + 72x + 32x2)' equal to zero and attempt to solve: Simplifying 39 + 72x + 32x2 = 0 Solving 39 + 72x + 32x2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. 1.21875 + 2.25x + x2 = 0 Move the constant term to the right: Add '-1.21875' to each side of the equation. 1.21875 + 2.25x + -1.21875 + x2 = 0 + -1.21875 Reorder the terms: 1.21875 + -1.21875 + 2.25x + x2 = 0 + -1.21875 Combine like terms: 1.21875 + -1.21875 = 0.00000 0.00000 + 2.25x + x2 = 0 + -1.21875 2.25x + x2 = 0 + -1.21875 Combine like terms: 0 + -1.21875 = -1.21875 2.25x + x2 = -1.21875 The x term is 2.25x. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25x + 1.265625 + x2 = -1.21875 + 1.265625 Reorder the terms: 1.265625 + 2.25x + x2 = -1.21875 + 1.265625 Combine like terms: -1.21875 + 1.265625 = 0.046875 1.265625 + 2.25x + x2 = 0.046875 Factor a perfect square on the left side: (x + 1.125)(x + 1.125) = 0.046875 Calculate the square root of the right side: 0.216506351 Break this problem into two subproblems by setting (x + 1.125) equal to 0.216506351 and -0.216506351.Subproblem 1
x + 1.125 = 0.216506351 Simplifying x + 1.125 = 0.216506351 Reorder the terms: 1.125 + x = 0.216506351 Solving 1.125 + x = 0.216506351 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = 0.216506351 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = 0.216506351 + -1.125 x = 0.216506351 + -1.125 Combine like terms: 0.216506351 + -1.125 = -0.908493649 x = -0.908493649 Simplifying x = -0.908493649Subproblem 2
x + 1.125 = -0.216506351 Simplifying x + 1.125 = -0.216506351 Reorder the terms: 1.125 + x = -0.216506351 Solving 1.125 + x = -0.216506351 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = -0.216506351 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = -0.216506351 + -1.125 x = -0.216506351 + -1.125 Combine like terms: -0.216506351 + -1.125 = -1.341506351 x = -1.341506351 Simplifying x = -1.341506351Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.908493649, -1.341506351}Solution
x = {-0.908493649, -1.341506351}
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