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Simplifying 18.897x2 + 64.509x + 4.9965 = 65 Reorder the terms: 4.9965 + 64.509x + 18.897x2 = 65 Solving 4.9965 + 64.509x + 18.897x2 = 65 Solving for variable 'x'. Reorder the terms: 4.9965 + -65 + 64.509x + 18.897x2 = 65 + -65 Combine like terms: 4.9965 + -65 = -60.0035 -60.0035 + 64.509x + 18.897x2 = 65 + -65 Combine like terms: 65 + -65 = 0 -60.0035 + 64.509x + 18.897x2 = 0 Begin completing the square. Divide all terms by 18.897 the coefficient of the squared term: Divide each side by '18.897'. -3.175292374 + 3.413716463x + x2 = 0 Move the constant term to the right: Add '3.175292374' to each side of the equation. -3.175292374 + 3.413716463x + 3.175292374 + x2 = 0 + 3.175292374 Reorder the terms: -3.175292374 + 3.175292374 + 3.413716463x + x2 = 0 + 3.175292374 Combine like terms: -3.175292374 + 3.175292374 = 0.000000000 0.000000000 + 3.413716463x + x2 = 0 + 3.175292374 3.413716463x + x2 = 0 + 3.175292374 Combine like terms: 0 + 3.175292374 = 3.175292374 3.413716463x + x2 = 3.175292374 The x term is 3.413716463x. Take half its coefficient (1.706858232). Square it (2.913365024) and add it to both sides. Add '2.913365024' to each side of the equation. 3.413716463x + 2.913365024 + x2 = 3.175292374 + 2.913365024 Reorder the terms: 2.913365024 + 3.413716463x + x2 = 3.175292374 + 2.913365024 Combine like terms: 3.175292374 + 2.913365024 = 6.088657398 2.913365024 + 3.413716463x + x2 = 6.088657398 Factor a perfect square on the left side: (x + 1.706858232)(x + 1.706858232) = 6.088657398 Calculate the square root of the right side: 2.467520496 Break this problem into two subproblems by setting (x + 1.706858232) equal to 2.467520496 and -2.467520496.Subproblem 1
x + 1.706858232 = 2.467520496 Simplifying x + 1.706858232 = 2.467520496 Reorder the terms: 1.706858232 + x = 2.467520496 Solving 1.706858232 + x = 2.467520496 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.706858232' to each side of the equation. 1.706858232 + -1.706858232 + x = 2.467520496 + -1.706858232 Combine like terms: 1.706858232 + -1.706858232 = 0.000000000 0.000000000 + x = 2.467520496 + -1.706858232 x = 2.467520496 + -1.706858232 Combine like terms: 2.467520496 + -1.706858232 = 0.760662264 x = 0.760662264 Simplifying x = 0.760662264Subproblem 2
x + 1.706858232 = -2.467520496 Simplifying x + 1.706858232 = -2.467520496 Reorder the terms: 1.706858232 + x = -2.467520496 Solving 1.706858232 + x = -2.467520496 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.706858232' to each side of the equation. 1.706858232 + -1.706858232 + x = -2.467520496 + -1.706858232 Combine like terms: 1.706858232 + -1.706858232 = 0.000000000 0.000000000 + x = -2.467520496 + -1.706858232 x = -2.467520496 + -1.706858232 Combine like terms: -2.467520496 + -1.706858232 = -4.174378728 x = -4.174378728 Simplifying x = -4.174378728Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.760662264, -4.174378728}
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