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Simplifying 0 = -16x2 + 32x + 40 Reorder the terms: 0 = 40 + 32x + -16x2 Solving 0 = 40 + 32x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -40 = -40 -40 + -32x + 16x2 = 40 + 32x + -16x2 + -40 + -32x + 16x2 Reorder the terms: -40 + -32x + 16x2 = 40 + -40 + 32x + -32x + -16x2 + 16x2 Combine like terms: 40 + -40 = 0 -40 + -32x + 16x2 = 0 + 32x + -32x + -16x2 + 16x2 -40 + -32x + 16x2 = 32x + -32x + -16x2 + 16x2 Combine like terms: 32x + -32x = 0 -40 + -32x + 16x2 = 0 + -16x2 + 16x2 -40 + -32x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -40 + -32x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-5 + -4x + 2x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-5 + -4x + 2x2)' equal to zero and attempt to solve: Simplifying -5 + -4x + 2x2 = 0 Solving -5 + -4x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + -2x + x2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -2x + 2.5 + x2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -2x + x2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -2x + x2 = 0 + 2.5 -2x + x2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -2x + x2 = 2.5 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = 2.5 + 1 Reorder the terms: 1 + -2x + x2 = 2.5 + 1 Combine like terms: 2.5 + 1 = 3.5 1 + -2x + x2 = 3.5 Factor a perfect square on the left side: (x + -1)(x + -1) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (x + -1) equal to 1.870828693 and -1.870828693.Subproblem 1
x + -1 = 1.870828693 Simplifying x + -1 = 1.870828693 Reorder the terms: -1 + x = 1.870828693 Solving -1 + x = 1.870828693 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 1.870828693 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.870828693 + 1 x = 1.870828693 + 1 Combine like terms: 1.870828693 + 1 = 2.870828693 x = 2.870828693 Simplifying x = 2.870828693Subproblem 2
x + -1 = -1.870828693 Simplifying x + -1 = -1.870828693 Reorder the terms: -1 + x = -1.870828693 Solving -1 + x = -1.870828693 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -1.870828693 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.870828693 + 1 x = -1.870828693 + 1 Combine like terms: -1.870828693 + 1 = -0.870828693 x = -0.870828693 Simplifying x = -0.870828693Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.870828693, -0.870828693}Solution
x = {2.870828693, -0.870828693}
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