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Simplifying -16x2 + 6x + 12 = 0 Reorder the terms: 12 + 6x + -16x2 = 0 Solving 12 + 6x + -16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(6 + 3x + -8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(6 + 3x + -8x2)' equal to zero and attempt to solve: Simplifying 6 + 3x + -8x2 = 0 Solving 6 + 3x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.75 + -0.375x + x2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.375x + 0.75 + x2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.375x + x2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.375x + x2 = 0 + 0.75 -0.375x + x2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -0.375x + x2 = 0.75 The x term is -0.375x. Take half its coefficient (-0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. -0.375x + 0.03515625 + x2 = 0.75 + 0.03515625 Reorder the terms: 0.03515625 + -0.375x + x2 = 0.75 + 0.03515625 Combine like terms: 0.75 + 0.03515625 = 0.78515625 0.03515625 + -0.375x + x2 = 0.78515625 Factor a perfect square on the left side: (x + -0.1875)(x + -0.1875) = 0.78515625 Calculate the square root of the right side: 0.88609043 Break this problem into two subproblems by setting (x + -0.1875) equal to 0.88609043 and -0.88609043.Subproblem 1
x + -0.1875 = 0.88609043 Simplifying x + -0.1875 = 0.88609043 Reorder the terms: -0.1875 + x = 0.88609043 Solving -0.1875 + x = 0.88609043 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + x = 0.88609043 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + x = 0.88609043 + 0.1875 x = 0.88609043 + 0.1875 Combine like terms: 0.88609043 + 0.1875 = 1.07359043 x = 1.07359043 Simplifying x = 1.07359043Subproblem 2
x + -0.1875 = -0.88609043 Simplifying x + -0.1875 = -0.88609043 Reorder the terms: -0.1875 + x = -0.88609043 Solving -0.1875 + x = -0.88609043 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + x = -0.88609043 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + x = -0.88609043 + 0.1875 x = -0.88609043 + 0.1875 Combine like terms: -0.88609043 + 0.1875 = -0.69859043 x = -0.69859043 Simplifying x = -0.69859043Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.07359043, -0.69859043}Solution
x = {1.07359043, -0.69859043}
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