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Simplifying a(12 + 4.9a) = 25 (12 * a + 4.9a * a) = 25 (12a + 4.9a2) = 25 Solving 12a + 4.9a2 = 25 Solving for variable 'a'. Reorder the terms: -25 + 12a + 4.9a2 = 25 + -25 Combine like terms: 25 + -25 = 0 -25 + 12a + 4.9a2 = 0 Begin completing the square. Divide all terms by 4.9 the coefficient of the squared term: Divide each side by '4.9'. -5.102040816 + 2.448979592a + a2 = 0 Move the constant term to the right: Add '5.102040816' to each side of the equation. -5.102040816 + 2.448979592a + 5.102040816 + a2 = 0 + 5.102040816 Reorder the terms: -5.102040816 + 5.102040816 + 2.448979592a + a2 = 0 + 5.102040816 Combine like terms: -5.102040816 + 5.102040816 = 0.000000000 0.000000000 + 2.448979592a + a2 = 0 + 5.102040816 2.448979592a + a2 = 0 + 5.102040816 Combine like terms: 0 + 5.102040816 = 5.102040816 2.448979592a + a2 = 5.102040816 The a term is 2.448979592a. Take half its coefficient (1.224489796). Square it (1.499375261) and add it to both sides. Add '1.499375261' to each side of the equation. 2.448979592a + 1.499375261 + a2 = 5.102040816 + 1.499375261 Reorder the terms: 1.499375261 + 2.448979592a + a2 = 5.102040816 + 1.499375261 Combine like terms: 5.102040816 + 1.499375261 = 6.601416077 1.499375261 + 2.448979592a + a2 = 6.601416077 Factor a perfect square on the left side: (a + 1.224489796)(a + 1.224489796) = 6.601416077 Calculate the square root of the right side: 2.569322105 Break this problem into two subproblems by setting (a + 1.224489796) equal to 2.569322105 and -2.569322105.Subproblem 1
a + 1.224489796 = 2.569322105 Simplifying a + 1.224489796 = 2.569322105 Reorder the terms: 1.224489796 + a = 2.569322105 Solving 1.224489796 + a = 2.569322105 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.224489796' to each side of the equation. 1.224489796 + -1.224489796 + a = 2.569322105 + -1.224489796 Combine like terms: 1.224489796 + -1.224489796 = 0.000000000 0.000000000 + a = 2.569322105 + -1.224489796 a = 2.569322105 + -1.224489796 Combine like terms: 2.569322105 + -1.224489796 = 1.344832309 a = 1.344832309 Simplifying a = 1.344832309Subproblem 2
a + 1.224489796 = -2.569322105 Simplifying a + 1.224489796 = -2.569322105 Reorder the terms: 1.224489796 + a = -2.569322105 Solving 1.224489796 + a = -2.569322105 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.224489796' to each side of the equation. 1.224489796 + -1.224489796 + a = -2.569322105 + -1.224489796 Combine like terms: 1.224489796 + -1.224489796 = 0.000000000 0.000000000 + a = -2.569322105 + -1.224489796 a = -2.569322105 + -1.224489796 Combine like terms: -2.569322105 + -1.224489796 = -3.793811901 a = -3.793811901 Simplifying a = -3.793811901Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.344832309, -3.793811901}
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