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Simplifying -16x2 + 62x + 23 = x Reorder the terms: 23 + 62x + -16x2 = x Solving 23 + 62x + -16x2 = x Solving for variable 'x'. Reorder the terms: 23 + 62x + -1x + -16x2 = x + -1x Combine like terms: 62x + -1x = 61x 23 + 61x + -16x2 = x + -1x Combine like terms: x + -1x = 0 23 + 61x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -1.4375 + -3.8125x + x2 = 0 Move the constant term to the right: Add '1.4375' to each side of the equation. -1.4375 + -3.8125x + 1.4375 + x2 = 0 + 1.4375 Reorder the terms: -1.4375 + 1.4375 + -3.8125x + x2 = 0 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + -3.8125x + x2 = 0 + 1.4375 -3.8125x + x2 = 0 + 1.4375 Combine like terms: 0 + 1.4375 = 1.4375 -3.8125x + x2 = 1.4375 The x term is -3.8125x. Take half its coefficient (-1.90625). Square it (3.633789063) and add it to both sides. Add '3.633789063' to each side of the equation. -3.8125x + 3.633789063 + x2 = 1.4375 + 3.633789063 Reorder the terms: 3.633789063 + -3.8125x + x2 = 1.4375 + 3.633789063 Combine like terms: 1.4375 + 3.633789063 = 5.071289063 3.633789063 + -3.8125x + x2 = 5.071289063 Factor a perfect square on the left side: (x + -1.90625)(x + -1.90625) = 5.071289063 Calculate the square root of the right side: 2.251952278 Break this problem into two subproblems by setting (x + -1.90625) equal to 2.251952278 and -2.251952278.Subproblem 1
x + -1.90625 = 2.251952278 Simplifying x + -1.90625 = 2.251952278 Reorder the terms: -1.90625 + x = 2.251952278 Solving -1.90625 + x = 2.251952278 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.90625' to each side of the equation. -1.90625 + 1.90625 + x = 2.251952278 + 1.90625 Combine like terms: -1.90625 + 1.90625 = 0.00000 0.00000 + x = 2.251952278 + 1.90625 x = 2.251952278 + 1.90625 Combine like terms: 2.251952278 + 1.90625 = 4.158202278 x = 4.158202278 Simplifying x = 4.158202278Subproblem 2
x + -1.90625 = -2.251952278 Simplifying x + -1.90625 = -2.251952278 Reorder the terms: -1.90625 + x = -2.251952278 Solving -1.90625 + x = -2.251952278 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.90625' to each side of the equation. -1.90625 + 1.90625 + x = -2.251952278 + 1.90625 Combine like terms: -1.90625 + 1.90625 = 0.00000 0.00000 + x = -2.251952278 + 1.90625 x = -2.251952278 + 1.90625 Combine like terms: -2.251952278 + 1.90625 = -0.345702278 x = -0.345702278 Simplifying x = -0.345702278Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.158202278, -0.345702278}
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