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Simplifying 0 = -16x2 + 102x + 455 Reorder the terms: 0 = 455 + 102x + -16x2 Solving 0 = 455 + 102x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -455 = -455 -455 + -102x + 16x2 = 455 + 102x + -16x2 + -455 + -102x + 16x2 Reorder the terms: -455 + -102x + 16x2 = 455 + -455 + 102x + -102x + -16x2 + 16x2 Combine like terms: 455 + -455 = 0 -455 + -102x + 16x2 = 0 + 102x + -102x + -16x2 + 16x2 -455 + -102x + 16x2 = 102x + -102x + -16x2 + 16x2 Combine like terms: 102x + -102x = 0 -455 + -102x + 16x2 = 0 + -16x2 + 16x2 -455 + -102x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -455 + -102x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -28.4375 + -6.375x + x2 = 0 Move the constant term to the right: Add '28.4375' to each side of the equation. -28.4375 + -6.375x + 28.4375 + x2 = 0 + 28.4375 Reorder the terms: -28.4375 + 28.4375 + -6.375x + x2 = 0 + 28.4375 Combine like terms: -28.4375 + 28.4375 = 0.0000 0.0000 + -6.375x + x2 = 0 + 28.4375 -6.375x + x2 = 0 + 28.4375 Combine like terms: 0 + 28.4375 = 28.4375 -6.375x + x2 = 28.4375 The x term is -6.375x. Take half its coefficient (-3.1875). Square it (10.16015625) and add it to both sides. Add '10.16015625' to each side of the equation. -6.375x + 10.16015625 + x2 = 28.4375 + 10.16015625 Reorder the terms: 10.16015625 + -6.375x + x2 = 28.4375 + 10.16015625 Combine like terms: 28.4375 + 10.16015625 = 38.59765625 10.16015625 + -6.375x + x2 = 38.59765625 Factor a perfect square on the left side: (x + -3.1875)(x + -3.1875) = 38.59765625 Calculate the square root of the right side: 6.212701204 Break this problem into two subproblems by setting (x + -3.1875) equal to 6.212701204 and -6.212701204.Subproblem 1
x + -3.1875 = 6.212701204 Simplifying x + -3.1875 = 6.212701204 Reorder the terms: -3.1875 + x = 6.212701204 Solving -3.1875 + x = 6.212701204 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.1875' to each side of the equation. -3.1875 + 3.1875 + x = 6.212701204 + 3.1875 Combine like terms: -3.1875 + 3.1875 = 0.0000 0.0000 + x = 6.212701204 + 3.1875 x = 6.212701204 + 3.1875 Combine like terms: 6.212701204 + 3.1875 = 9.400201204 x = 9.400201204 Simplifying x = 9.400201204Subproblem 2
x + -3.1875 = -6.212701204 Simplifying x + -3.1875 = -6.212701204 Reorder the terms: -3.1875 + x = -6.212701204 Solving -3.1875 + x = -6.212701204 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.1875' to each side of the equation. -3.1875 + 3.1875 + x = -6.212701204 + 3.1875 Combine like terms: -3.1875 + 3.1875 = 0.0000 0.0000 + x = -6.212701204 + 3.1875 x = -6.212701204 + 3.1875 Combine like terms: -6.212701204 + 3.1875 = -3.025201204 x = -3.025201204 Simplifying x = -3.025201204Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.400201204, -3.025201204}
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