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Simplifying -16x2 + 78x + 6 = 0 Reorder the terms: 6 + 78x + -16x2 = 0 Solving 6 + 78x + -16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(3 + 39x + -8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(3 + 39x + -8x2)' equal to zero and attempt to solve: Simplifying 3 + 39x + -8x2 = 0 Solving 3 + 39x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.375 + -4.875x + x2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -4.875x + 0.375 + x2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -4.875x + x2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -4.875x + x2 = 0 + 0.375 -4.875x + x2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -4.875x + x2 = 0.375 The x term is -4.875x. Take half its coefficient (-2.4375). Square it (5.94140625) and add it to both sides. Add '5.94140625' to each side of the equation. -4.875x + 5.94140625 + x2 = 0.375 + 5.94140625 Reorder the terms: 5.94140625 + -4.875x + x2 = 0.375 + 5.94140625 Combine like terms: 0.375 + 5.94140625 = 6.31640625 5.94140625 + -4.875x + x2 = 6.31640625 Factor a perfect square on the left side: (x + -2.4375)(x + -2.4375) = 6.31640625 Calculate the square root of the right side: 2.513246158 Break this problem into two subproblems by setting (x + -2.4375) equal to 2.513246158 and -2.513246158.Subproblem 1
x + -2.4375 = 2.513246158 Simplifying x + -2.4375 = 2.513246158 Reorder the terms: -2.4375 + x = 2.513246158 Solving -2.4375 + x = 2.513246158 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.4375' to each side of the equation. -2.4375 + 2.4375 + x = 2.513246158 + 2.4375 Combine like terms: -2.4375 + 2.4375 = 0.0000 0.0000 + x = 2.513246158 + 2.4375 x = 2.513246158 + 2.4375 Combine like terms: 2.513246158 + 2.4375 = 4.950746158 x = 4.950746158 Simplifying x = 4.950746158Subproblem 2
x + -2.4375 = -2.513246158 Simplifying x + -2.4375 = -2.513246158 Reorder the terms: -2.4375 + x = -2.513246158 Solving -2.4375 + x = -2.513246158 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.4375' to each side of the equation. -2.4375 + 2.4375 + x = -2.513246158 + 2.4375 Combine like terms: -2.4375 + 2.4375 = 0.0000 0.0000 + x = -2.513246158 + 2.4375 x = -2.513246158 + 2.4375 Combine like terms: -2.513246158 + 2.4375 = -0.075746158 x = -0.075746158 Simplifying x = -0.075746158Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.950746158, -0.075746158}Solution
x = {4.950746158, -0.075746158}
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