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Simplifying 9a2 + -24a = -13 Reorder the terms: -24a + 9a2 = -13 Solving -24a + 9a2 = -13 Solving for variable 'a'. Reorder the terms: 13 + -24a + 9a2 = -13 + 13 Combine like terms: -13 + 13 = 0 13 + -24a + 9a2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 1.444444444 + -2.666666667a + a2 = 0 Move the constant term to the right: Add '-1.444444444' to each side of the equation. 1.444444444 + -2.666666667a + -1.444444444 + a2 = 0 + -1.444444444 Reorder the terms: 1.444444444 + -1.444444444 + -2.666666667a + a2 = 0 + -1.444444444 Combine like terms: 1.444444444 + -1.444444444 = 0.000000000 0.000000000 + -2.666666667a + a2 = 0 + -1.444444444 -2.666666667a + a2 = 0 + -1.444444444 Combine like terms: 0 + -1.444444444 = -1.444444444 -2.666666667a + a2 = -1.444444444 The a term is -2.666666667a. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667a + 1.777777780 + a2 = -1.444444444 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667a + a2 = -1.444444444 + 1.777777780 Combine like terms: -1.444444444 + 1.777777780 = 0.333333336 1.777777780 + -2.666666667a + a2 = 0.333333336 Factor a perfect square on the left side: (a + -1.333333334)(a + -1.333333334) = 0.333333336 Calculate the square root of the right side: 0.577350271 Break this problem into two subproblems by setting (a + -1.333333334) equal to 0.577350271 and -0.577350271.Subproblem 1
a + -1.333333334 = 0.577350271 Simplifying a + -1.333333334 = 0.577350271 Reorder the terms: -1.333333334 + a = 0.577350271 Solving -1.333333334 + a = 0.577350271 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + a = 0.577350271 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + a = 0.577350271 + 1.333333334 a = 0.577350271 + 1.333333334 Combine like terms: 0.577350271 + 1.333333334 = 1.910683605 a = 1.910683605 Simplifying a = 1.910683605Subproblem 2
a + -1.333333334 = -0.577350271 Simplifying a + -1.333333334 = -0.577350271 Reorder the terms: -1.333333334 + a = -0.577350271 Solving -1.333333334 + a = -0.577350271 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + a = -0.577350271 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + a = -0.577350271 + 1.333333334 a = -0.577350271 + 1.333333334 Combine like terms: -0.577350271 + 1.333333334 = 0.755983063 a = 0.755983063 Simplifying a = 0.755983063Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.910683605, 0.755983063}
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