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Simplifying 9a2 = 30a + 25 Reorder the terms: 9a2 = 25 + 30a Solving 9a2 = 25 + 30a Solving for variable 'a'. Reorder the terms: -25 + -30a + 9a2 = 25 + 30a + -25 + -30a Reorder the terms: -25 + -30a + 9a2 = 25 + -25 + 30a + -30a Combine like terms: 25 + -25 = 0 -25 + -30a + 9a2 = 0 + 30a + -30a -25 + -30a + 9a2 = 30a + -30a Combine like terms: 30a + -30a = 0 -25 + -30a + 9a2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.777777778 + -3.333333333a + a2 = 0 Move the constant term to the right: Add '2.777777778' to each side of the equation. -2.777777778 + -3.333333333a + 2.777777778 + a2 = 0 + 2.777777778 Reorder the terms: -2.777777778 + 2.777777778 + -3.333333333a + a2 = 0 + 2.777777778 Combine like terms: -2.777777778 + 2.777777778 = 0.000000000 0.000000000 + -3.333333333a + a2 = 0 + 2.777777778 -3.333333333a + a2 = 0 + 2.777777778 Combine like terms: 0 + 2.777777778 = 2.777777778 -3.333333333a + a2 = 2.777777778 The a term is -3.333333333a. Take half its coefficient (-1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. -3.333333333a + 2.777777779 + a2 = 2.777777778 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333a + a2 = 2.777777778 + 2.777777779 Combine like terms: 2.777777778 + 2.777777779 = 5.555555557 2.777777779 + -3.333333333a + a2 = 5.555555557 Factor a perfect square on the left side: (a + -1.666666667)(a + -1.666666667) = 5.555555557 Calculate the square root of the right side: 2.357022604 Break this problem into two subproblems by setting (a + -1.666666667) equal to 2.357022604 and -2.357022604.Subproblem 1
a + -1.666666667 = 2.357022604 Simplifying a + -1.666666667 = 2.357022604 Reorder the terms: -1.666666667 + a = 2.357022604 Solving -1.666666667 + a = 2.357022604 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + a = 2.357022604 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + a = 2.357022604 + 1.666666667 a = 2.357022604 + 1.666666667 Combine like terms: 2.357022604 + 1.666666667 = 4.023689271 a = 4.023689271 Simplifying a = 4.023689271Subproblem 2
a + -1.666666667 = -2.357022604 Simplifying a + -1.666666667 = -2.357022604 Reorder the terms: -1.666666667 + a = -2.357022604 Solving -1.666666667 + a = -2.357022604 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + a = -2.357022604 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + a = -2.357022604 + 1.666666667 a = -2.357022604 + 1.666666667 Combine like terms: -2.357022604 + 1.666666667 = -0.690355937 a = -0.690355937 Simplifying a = -0.690355937Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.023689271, -0.690355937}
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