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Simplifying 3a(2a + -5) = 130 + -2(a + 12) Reorder the terms: 3a(-5 + 2a) = 130 + -2(a + 12) (-5 * 3a + 2a * 3a) = 130 + -2(a + 12) (-15a + 6a2) = 130 + -2(a + 12) Reorder the terms: -15a + 6a2 = 130 + -2(12 + a) -15a + 6a2 = 130 + (12 * -2 + a * -2) -15a + 6a2 = 130 + (-24 + -2a) Combine like terms: 130 + -24 = 106 -15a + 6a2 = 106 + -2a Solving -15a + 6a2 = 106 + -2a Solving for variable 'a'. Reorder the terms: -106 + -15a + 2a + 6a2 = 106 + -2a + -106 + 2a Combine like terms: -15a + 2a = -13a -106 + -13a + 6a2 = 106 + -2a + -106 + 2a Reorder the terms: -106 + -13a + 6a2 = 106 + -106 + -2a + 2a Combine like terms: 106 + -106 = 0 -106 + -13a + 6a2 = 0 + -2a + 2a -106 + -13a + 6a2 = -2a + 2a Combine like terms: -2a + 2a = 0 -106 + -13a + 6a2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -17.66666667 + -2.166666667a + a2 = 0 Move the constant term to the right: Add '17.66666667' to each side of the equation. -17.66666667 + -2.166666667a + 17.66666667 + a2 = 0 + 17.66666667 Reorder the terms: -17.66666667 + 17.66666667 + -2.166666667a + a2 = 0 + 17.66666667 Combine like terms: -17.66666667 + 17.66666667 = 0.00000000 0.00000000 + -2.166666667a + a2 = 0 + 17.66666667 -2.166666667a + a2 = 0 + 17.66666667 Combine like terms: 0 + 17.66666667 = 17.66666667 -2.166666667a + a2 = 17.66666667 The a term is -2.166666667a. Take half its coefficient (-1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. -2.166666667a + 1.173611113 + a2 = 17.66666667 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667a + a2 = 17.66666667 + 1.173611113 Combine like terms: 17.66666667 + 1.173611113 = 18.840277783 1.173611113 + -2.166666667a + a2 = 18.840277783 Factor a perfect square on the left side: (a + -1.083333334)(a + -1.083333334) = 18.840277783 Calculate the square root of the right side: 4.340538882 Break this problem into two subproblems by setting (a + -1.083333334) equal to 4.340538882 and -4.340538882.Subproblem 1
a + -1.083333334 = 4.340538882 Simplifying a + -1.083333334 = 4.340538882 Reorder the terms: -1.083333334 + a = 4.340538882 Solving -1.083333334 + a = 4.340538882 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + a = 4.340538882 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + a = 4.340538882 + 1.083333334 a = 4.340538882 + 1.083333334 Combine like terms: 4.340538882 + 1.083333334 = 5.423872216 a = 5.423872216 Simplifying a = 5.423872216Subproblem 2
a + -1.083333334 = -4.340538882 Simplifying a + -1.083333334 = -4.340538882 Reorder the terms: -1.083333334 + a = -4.340538882 Solving -1.083333334 + a = -4.340538882 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + a = -4.340538882 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + a = -4.340538882 + 1.083333334 a = -4.340538882 + 1.083333334 Combine like terms: -4.340538882 + 1.083333334 = -3.257205548 a = -3.257205548 Simplifying a = -3.257205548Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.423872216, -3.257205548}
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