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Simplifying 5(3a + -6) * 5a = -12 + a Reorder the terms: 5(-6 + 3a) * 5a = -12 + a Reorder the terms for easier multiplication: 5 * 5a(-6 + 3a) = -12 + a Multiply 5 * 5 25a(-6 + 3a) = -12 + a (-6 * 25a + 3a * 25a) = -12 + a (-150a + 75a2) = -12 + a Solving -150a + 75a2 = -12 + a Solving for variable 'a'. Reorder the terms: 12 + -150a + -1a + 75a2 = -12 + a + 12 + -1a Combine like terms: -150a + -1a = -151a 12 + -151a + 75a2 = -12 + a + 12 + -1a Reorder the terms: 12 + -151a + 75a2 = -12 + 12 + a + -1a Combine like terms: -12 + 12 = 0 12 + -151a + 75a2 = 0 + a + -1a 12 + -151a + 75a2 = a + -1a Combine like terms: a + -1a = 0 12 + -151a + 75a2 = 0 Begin completing the square. Divide all terms by 75 the coefficient of the squared term: Divide each side by '75'. 0.16 + -2.013333333a + a2 = 0 Move the constant term to the right: Add '-0.16' to each side of the equation. 0.16 + -2.013333333a + -0.16 + a2 = 0 + -0.16 Reorder the terms: 0.16 + -0.16 + -2.013333333a + a2 = 0 + -0.16 Combine like terms: 0.16 + -0.16 = 0.00 0.00 + -2.013333333a + a2 = 0 + -0.16 -2.013333333a + a2 = 0 + -0.16 Combine like terms: 0 + -0.16 = -0.16 -2.013333333a + a2 = -0.16 The a term is -2.013333333a. Take half its coefficient (-1.006666667). Square it (1.013377778) and add it to both sides. Add '1.013377778' to each side of the equation. -2.013333333a + 1.013377778 + a2 = -0.16 + 1.013377778 Reorder the terms: 1.013377778 + -2.013333333a + a2 = -0.16 + 1.013377778 Combine like terms: -0.16 + 1.013377778 = 0.853377778 1.013377778 + -2.013333333a + a2 = 0.853377778 Factor a perfect square on the left side: (a + -1.006666667)(a + -1.006666667) = 0.853377778 Calculate the square root of the right side: 0.923784487 Break this problem into two subproblems by setting (a + -1.006666667) equal to 0.923784487 and -0.923784487.Subproblem 1
a + -1.006666667 = 0.923784487 Simplifying a + -1.006666667 = 0.923784487 Reorder the terms: -1.006666667 + a = 0.923784487 Solving -1.006666667 + a = 0.923784487 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.006666667' to each side of the equation. -1.006666667 + 1.006666667 + a = 0.923784487 + 1.006666667 Combine like terms: -1.006666667 + 1.006666667 = 0.000000000 0.000000000 + a = 0.923784487 + 1.006666667 a = 0.923784487 + 1.006666667 Combine like terms: 0.923784487 + 1.006666667 = 1.930451154 a = 1.930451154 Simplifying a = 1.930451154Subproblem 2
a + -1.006666667 = -0.923784487 Simplifying a + -1.006666667 = -0.923784487 Reorder the terms: -1.006666667 + a = -0.923784487 Solving -1.006666667 + a = -0.923784487 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.006666667' to each side of the equation. -1.006666667 + 1.006666667 + a = -0.923784487 + 1.006666667 Combine like terms: -1.006666667 + 1.006666667 = 0.000000000 0.000000000 + a = -0.923784487 + 1.006666667 a = -0.923784487 + 1.006666667 Combine like terms: -0.923784487 + 1.006666667 = 0.08288218 a = 0.08288218 Simplifying a = 0.08288218Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.930451154, 0.08288218}
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