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Simplifying 0 = -16x2 + 14x + 40 Reorder the terms: 0 = 40 + 14x + -16x2 Solving 0 = 40 + 14x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -40 = -40 -40 + -14x + 16x2 = 40 + 14x + -16x2 + -40 + -14x + 16x2 Reorder the terms: -40 + -14x + 16x2 = 40 + -40 + 14x + -14x + -16x2 + 16x2 Combine like terms: 40 + -40 = 0 -40 + -14x + 16x2 = 0 + 14x + -14x + -16x2 + 16x2 -40 + -14x + 16x2 = 14x + -14x + -16x2 + 16x2 Combine like terms: 14x + -14x = 0 -40 + -14x + 16x2 = 0 + -16x2 + 16x2 -40 + -14x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -40 + -14x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-20 + -7x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-20 + -7x + 8x2)' equal to zero and attempt to solve: Simplifying -20 + -7x + 8x2 = 0 Solving -20 + -7x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.5 + -0.875x + x2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -0.875x + 2.5 + x2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -0.875x + x2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -0.875x + x2 = 0 + 2.5 -0.875x + x2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -0.875x + x2 = 2.5 The x term is -0.875x. Take half its coefficient (-0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. -0.875x + 0.19140625 + x2 = 2.5 + 0.19140625 Reorder the terms: 0.19140625 + -0.875x + x2 = 2.5 + 0.19140625 Combine like terms: 2.5 + 0.19140625 = 2.69140625 0.19140625 + -0.875x + x2 = 2.69140625 Factor a perfect square on the left side: (x + -0.4375)(x + -0.4375) = 2.69140625 Calculate the square root of the right side: 1.640550594 Break this problem into two subproblems by setting (x + -0.4375) equal to 1.640550594 and -1.640550594.Subproblem 1
x + -0.4375 = 1.640550594 Simplifying x + -0.4375 = 1.640550594 Reorder the terms: -0.4375 + x = 1.640550594 Solving -0.4375 + x = 1.640550594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + x = 1.640550594 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + x = 1.640550594 + 0.4375 x = 1.640550594 + 0.4375 Combine like terms: 1.640550594 + 0.4375 = 2.078050594 x = 2.078050594 Simplifying x = 2.078050594Subproblem 2
x + -0.4375 = -1.640550594 Simplifying x + -0.4375 = -1.640550594 Reorder the terms: -0.4375 + x = -1.640550594 Solving -0.4375 + x = -1.640550594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + x = -1.640550594 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + x = -1.640550594 + 0.4375 x = -1.640550594 + 0.4375 Combine like terms: -1.640550594 + 0.4375 = -1.203050594 x = -1.203050594 Simplifying x = -1.203050594Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.078050594, -1.203050594}Solution
x = {2.078050594, -1.203050594}
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