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Simplifying -16x2 + 30x + 20 = 0 Reorder the terms: 20 + 30x + -16x2 = 0 Solving 20 + 30x + -16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(10 + 15x + -8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(10 + 15x + -8x2)' equal to zero and attempt to solve: Simplifying 10 + 15x + -8x2 = 0 Solving 10 + 15x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1.25 + -1.875x + x2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -1.875x + 1.25 + x2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -1.875x + x2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -1.875x + x2 = 0 + 1.25 -1.875x + x2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -1.875x + x2 = 1.25 The x term is -1.875x. Take half its coefficient (-0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. -1.875x + 0.87890625 + x2 = 1.25 + 0.87890625 Reorder the terms: 0.87890625 + -1.875x + x2 = 1.25 + 0.87890625 Combine like terms: 1.25 + 0.87890625 = 2.12890625 0.87890625 + -1.875x + x2 = 2.12890625 Factor a perfect square on the left side: (x + -0.9375)(x + -0.9375) = 2.12890625 Calculate the square root of the right side: 1.459077191 Break this problem into two subproblems by setting (x + -0.9375) equal to 1.459077191 and -1.459077191.Subproblem 1
x + -0.9375 = 1.459077191 Simplifying x + -0.9375 = 1.459077191 Reorder the terms: -0.9375 + x = 1.459077191 Solving -0.9375 + x = 1.459077191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + x = 1.459077191 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + x = 1.459077191 + 0.9375 x = 1.459077191 + 0.9375 Combine like terms: 1.459077191 + 0.9375 = 2.396577191 x = 2.396577191 Simplifying x = 2.396577191Subproblem 2
x + -0.9375 = -1.459077191 Simplifying x + -0.9375 = -1.459077191 Reorder the terms: -0.9375 + x = -1.459077191 Solving -0.9375 + x = -1.459077191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + x = -1.459077191 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + x = -1.459077191 + 0.9375 x = -1.459077191 + 0.9375 Combine like terms: -1.459077191 + 0.9375 = -0.521577191 x = -0.521577191 Simplifying x = -0.521577191Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.396577191, -0.521577191}Solution
x = {2.396577191, -0.521577191}
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