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Simplifying 3a(2a + -5) = 13 + -2(a + 2) Reorder the terms: 3a(-5 + 2a) = 13 + -2(a + 2) (-5 * 3a + 2a * 3a) = 13 + -2(a + 2) (-15a + 6a2) = 13 + -2(a + 2) Reorder the terms: -15a + 6a2 = 13 + -2(2 + a) -15a + 6a2 = 13 + (2 * -2 + a * -2) -15a + 6a2 = 13 + (-4 + -2a) Combine like terms: 13 + -4 = 9 -15a + 6a2 = 9 + -2a Solving -15a + 6a2 = 9 + -2a Solving for variable 'a'. Reorder the terms: -9 + -15a + 2a + 6a2 = 9 + -2a + -9 + 2a Combine like terms: -15a + 2a = -13a -9 + -13a + 6a2 = 9 + -2a + -9 + 2a Reorder the terms: -9 + -13a + 6a2 = 9 + -9 + -2a + 2a Combine like terms: 9 + -9 = 0 -9 + -13a + 6a2 = 0 + -2a + 2a -9 + -13a + 6a2 = -2a + 2a Combine like terms: -2a + 2a = 0 -9 + -13a + 6a2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.5 + -2.166666667a + a2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -2.166666667a + 1.5 + a2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -2.166666667a + a2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -2.166666667a + a2 = 0 + 1.5 -2.166666667a + a2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -2.166666667a + a2 = 1.5 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 = 1.5 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667a + a2 = 1.5 + 1.173611113 Combine like terms: 1.5 + 1.173611113 = 2.673611113 1.173611113 + -2.166666667a + a2 = 2.673611113 Factor a perfect square on the left side: (a + -1.083333334)(a + -1.083333334) = 2.673611113 Calculate the square root of the right side: 1.635118073 Break this problem into two subproblems by setting (a + -1.083333334) equal to 1.635118073 and -1.635118073.Subproblem 1
a + -1.083333334 = 1.635118073 Simplifying a + -1.083333334 = 1.635118073 Reorder the terms: -1.083333334 + a = 1.635118073 Solving -1.083333334 + a = 1.635118073 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 = 1.635118073 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + a = 1.635118073 + 1.083333334 a = 1.635118073 + 1.083333334 Combine like terms: 1.635118073 + 1.083333334 = 2.718451407 a = 2.718451407 Simplifying a = 2.718451407Subproblem 2
a + -1.083333334 = -1.635118073 Simplifying a + -1.083333334 = -1.635118073 Reorder the terms: -1.083333334 + a = -1.635118073 Solving -1.083333334 + a = -1.635118073 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 = -1.635118073 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + a = -1.635118073 + 1.083333334 a = -1.635118073 + 1.083333334 Combine like terms: -1.635118073 + 1.083333334 = -0.551784739 a = -0.551784739 Simplifying a = -0.551784739Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.718451407, -0.551784739}
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