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Simplifying 25.1 = 0.14X2 + -0.78X + 5.12 Reorder the terms: 25.1 = 5.12 + -0.78X + 0.14X2 Solving 25.1 = 5.12 + -0.78X + 0.14X2 Solving for variable 'X'. Combine like terms: 25.1 + -5.12 = 19.98 19.98 + 0.78X + -0.14X2 = 5.12 + -0.78X + 0.14X2 + -5.12 + 0.78X + -0.14X2 Reorder the terms: 19.98 + 0.78X + -0.14X2 = 5.12 + -5.12 + -0.78X + 0.78X + 0.14X2 + -0.14X2 Combine like terms: 5.12 + -5.12 = 0.00 19.98 + 0.78X + -0.14X2 = 0.00 + -0.78X + 0.78X + 0.14X2 + -0.14X2 19.98 + 0.78X + -0.14X2 = -0.78X + 0.78X + 0.14X2 + -0.14X2 Combine like terms: -0.78X + 0.78X = 0.00 19.98 + 0.78X + -0.14X2 = 0.00 + 0.14X2 + -0.14X2 19.98 + 0.78X + -0.14X2 = 0.14X2 + -0.14X2 Combine like terms: 0.14X2 + -0.14X2 = 0.00 19.98 + 0.78X + -0.14X2 = 0.00 Begin completing the square. Divide all terms by -0.14 the coefficient of the squared term: Divide each side by '-0.14'. -142.7142857 + -5.571428571X + X2 = 0 Move the constant term to the right: Add '142.7142857' to each side of the equation. -142.7142857 + -5.571428571X + 142.7142857 + X2 = 0 + 142.7142857 Reorder the terms: -142.7142857 + 142.7142857 + -5.571428571X + X2 = 0 + 142.7142857 Combine like terms: -142.7142857 + 142.7142857 = 0.0000000 0.0000000 + -5.571428571X + X2 = 0 + 142.7142857 -5.571428571X + X2 = 0 + 142.7142857 Combine like terms: 0 + 142.7142857 = 142.7142857 -5.571428571X + X2 = 142.7142857 The X term is -5.571428571X. Take half its coefficient (-2.785714286). Square it (7.760204083) and add it to both sides. Add '7.760204083' to each side of the equation. -5.571428571X + 7.760204083 + X2 = 142.7142857 + 7.760204083 Reorder the terms: 7.760204083 + -5.571428571X + X2 = 142.7142857 + 7.760204083 Combine like terms: 142.7142857 + 7.760204083 = 150.474489783 7.760204083 + -5.571428571X + X2 = 150.474489783 Factor a perfect square on the left side: (X + -2.785714286)(X + -2.785714286) = 150.474489783 Calculate the square root of the right side: 12.266804383 Break this problem into two subproblems by setting (X + -2.785714286) equal to 12.266804383 and -12.266804383.Subproblem 1
X + -2.785714286 = 12.266804383 Simplifying X + -2.785714286 = 12.266804383 Reorder the terms: -2.785714286 + X = 12.266804383 Solving -2.785714286 + X = 12.266804383 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '2.785714286' to each side of the equation. -2.785714286 + 2.785714286 + X = 12.266804383 + 2.785714286 Combine like terms: -2.785714286 + 2.785714286 = 0.000000000 0.000000000 + X = 12.266804383 + 2.785714286 X = 12.266804383 + 2.785714286 Combine like terms: 12.266804383 + 2.785714286 = 15.052518669 X = 15.052518669 Simplifying X = 15.052518669Subproblem 2
X + -2.785714286 = -12.266804383 Simplifying X + -2.785714286 = -12.266804383 Reorder the terms: -2.785714286 + X = -12.266804383 Solving -2.785714286 + X = -12.266804383 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '2.785714286' to each side of the equation. -2.785714286 + 2.785714286 + X = -12.266804383 + 2.785714286 Combine like terms: -2.785714286 + 2.785714286 = 0.000000000 0.000000000 + X = -12.266804383 + 2.785714286 X = -12.266804383 + 2.785714286 Combine like terms: -12.266804383 + 2.785714286 = -9.481090097 X = -9.481090097 Simplifying X = -9.481090097Solution
The solution to the problem is based on the solutions from the subproblems. X = {15.052518669, -9.481090097}
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