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Simplifying 7 = 0.5(6 + 2a) * 3a Reorder the terms for easier multiplication: 7 = 0.5 * 3a(6 + 2a) Multiply 0.5 * 3 7 = 1.5a(6 + 2a) 7 = (6 * 1.5a + 2a * 1.5a) 7 = (9a + 3a2) Solving 7 = 9a + 3a2 Solving for variable 'a'. Reorder the terms: 7 + -9a + -3a2 = 9a + -9a + 3a2 + -3a2 Combine like terms: 9a + -9a = 0 7 + -9a + -3a2 = 0 + 3a2 + -3a2 7 + -9a + -3a2 = 3a2 + -3a2 Combine like terms: 3a2 + -3a2 = 0 7 + -9a + -3a2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -2.333333333 + 3a + a2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + 3a + 2.333333333 + a2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + 3a + a2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + 3a + a2 = 0 + 2.333333333 3a + a2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 3a + a2 = 2.333333333 The a term is 3a. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3a + 2.25 + a2 = 2.333333333 + 2.25 Reorder the terms: 2.25 + 3a + a2 = 2.333333333 + 2.25 Combine like terms: 2.333333333 + 2.25 = 4.583333333 2.25 + 3a + a2 = 4.583333333 Factor a perfect square on the left side: (a + 1.5)(a + 1.5) = 4.583333333 Calculate the square root of the right side: 2.140872096 Break this problem into two subproblems by setting (a + 1.5) equal to 2.140872096 and -2.140872096.Subproblem 1
a + 1.5 = 2.140872096 Simplifying a + 1.5 = 2.140872096 Reorder the terms: 1.5 + a = 2.140872096 Solving 1.5 + a = 2.140872096 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 2.140872096 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 2.140872096 + -1.5 a = 2.140872096 + -1.5 Combine like terms: 2.140872096 + -1.5 = 0.640872096 a = 0.640872096 Simplifying a = 0.640872096Subproblem 2
a + 1.5 = -2.140872096 Simplifying a + 1.5 = -2.140872096 Reorder the terms: 1.5 + a = -2.140872096 Solving 1.5 + a = -2.140872096 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -2.140872096 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -2.140872096 + -1.5 a = -2.140872096 + -1.5 Combine like terms: -2.140872096 + -1.5 = -3.640872096 a = -3.640872096 Simplifying a = -3.640872096Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.640872096, -3.640872096}
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