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Simplifying 2a(5a + -13) = 47 Reorder the terms: 2a(-13 + 5a) = 47 (-13 * 2a + 5a * 2a) = 47 (-26a + 10a2) = 47 Solving -26a + 10a2 = 47 Solving for variable 'a'. Reorder the terms: -47 + -26a + 10a2 = 47 + -47 Combine like terms: 47 + -47 = 0 -47 + -26a + 10a2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -4.7 + -2.6a + a2 = 0.0 Move the constant term to the right: Add '4.7' to each side of the equation. -4.7 + -2.6a + 4.7 + a2 = 0.0 + 4.7 Reorder the terms: -4.7 + 4.7 + -2.6a + a2 = 0.0 + 4.7 Combine like terms: -4.7 + 4.7 = 0.0 0.0 + -2.6a + a2 = 0.0 + 4.7 -2.6a + a2 = 0.0 + 4.7 Combine like terms: 0.0 + 4.7 = 4.7 -2.6a + a2 = 4.7 The a term is -2.6a. Take half its coefficient (-1.3). Square it (1.69) and add it to both sides. Add '1.69' to each side of the equation. -2.6a + 1.69 + a2 = 4.7 + 1.69 Reorder the terms: 1.69 + -2.6a + a2 = 4.7 + 1.69 Combine like terms: 4.7 + 1.69 = 6.39 1.69 + -2.6a + a2 = 6.39 Factor a perfect square on the left side: (a + -1.3)(a + -1.3) = 6.39 Calculate the square root of the right side: 2.527844932 Break this problem into two subproblems by setting (a + -1.3) equal to 2.527844932 and -2.527844932.Subproblem 1
a + -1.3 = 2.527844932 Simplifying a + -1.3 = 2.527844932 Reorder the terms: -1.3 + a = 2.527844932 Solving -1.3 + a = 2.527844932 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.3' to each side of the equation. -1.3 + 1.3 + a = 2.527844932 + 1.3 Combine like terms: -1.3 + 1.3 = 0.0 0.0 + a = 2.527844932 + 1.3 a = 2.527844932 + 1.3 Combine like terms: 2.527844932 + 1.3 = 3.827844932 a = 3.827844932 Simplifying a = 3.827844932Subproblem 2
a + -1.3 = -2.527844932 Simplifying a + -1.3 = -2.527844932 Reorder the terms: -1.3 + a = -2.527844932 Solving -1.3 + a = -2.527844932 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.3' to each side of the equation. -1.3 + 1.3 + a = -2.527844932 + 1.3 Combine like terms: -1.3 + 1.3 = 0.0 0.0 + a = -2.527844932 + 1.3 a = -2.527844932 + 1.3 Combine like terms: -2.527844932 + 1.3 = -1.227844932 a = -1.227844932 Simplifying a = -1.227844932Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.827844932, -1.227844932}
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