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Simplifying 2a(5a + 7) = 19 Reorder the terms: 2a(7 + 5a) = 19 (7 * 2a + 5a * 2a) = 19 (14a + 10a2) = 19 Solving 14a + 10a2 = 19 Solving for variable 'a'. Reorder the terms: -19 + 14a + 10a2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + 14a + 10a2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -1.9 + 1.4a + a2 = 0.0 Move the constant term to the right: Add '1.9' to each side of the equation. -1.9 + 1.4a + 1.9 + a2 = 0.0 + 1.9 Reorder the terms: -1.9 + 1.9 + 1.4a + a2 = 0.0 + 1.9 Combine like terms: -1.9 + 1.9 = 0.0 0.0 + 1.4a + a2 = 0.0 + 1.9 1.4a + a2 = 0.0 + 1.9 Combine like terms: 0.0 + 1.9 = 1.9 1.4a + a2 = 1.9 The a term is 1.4a. Take half its coefficient (0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. 1.4a + 0.49 + a2 = 1.9 + 0.49 Reorder the terms: 0.49 + 1.4a + a2 = 1.9 + 0.49 Combine like terms: 1.9 + 0.49 = 2.39 0.49 + 1.4a + a2 = 2.39 Factor a perfect square on the left side: (a + 0.7)(a + 0.7) = 2.39 Calculate the square root of the right side: 1.545962483 Break this problem into two subproblems by setting (a + 0.7) equal to 1.545962483 and -1.545962483.Subproblem 1
a + 0.7 = 1.545962483 Simplifying a + 0.7 = 1.545962483 Reorder the terms: 0.7 + a = 1.545962483 Solving 0.7 + a = 1.545962483 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + a = 1.545962483 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + a = 1.545962483 + -0.7 a = 1.545962483 + -0.7 Combine like terms: 1.545962483 + -0.7 = 0.845962483 a = 0.845962483 Simplifying a = 0.845962483Subproblem 2
a + 0.7 = -1.545962483 Simplifying a + 0.7 = -1.545962483 Reorder the terms: 0.7 + a = -1.545962483 Solving 0.7 + a = -1.545962483 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + a = -1.545962483 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + a = -1.545962483 + -0.7 a = -1.545962483 + -0.7 Combine like terms: -1.545962483 + -0.7 = -2.245962483 a = -2.245962483 Simplifying a = -2.245962483Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.845962483, -2.245962483}
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