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Simplifying (2a + 3)(4a + -8) = 7 Reorder the terms: (3 + 2a)(4a + -8) = 7 Reorder the terms: (3 + 2a)(-8 + 4a) = 7 Multiply (3 + 2a) * (-8 + 4a) (3(-8 + 4a) + 2a * (-8 + 4a)) = 7 ((-8 * 3 + 4a * 3) + 2a * (-8 + 4a)) = 7 ((-24 + 12a) + 2a * (-8 + 4a)) = 7 (-24 + 12a + (-8 * 2a + 4a * 2a)) = 7 (-24 + 12a + (-16a + 8a2)) = 7 Combine like terms: 12a + -16a = -4a (-24 + -4a + 8a2) = 7 Solving -24 + -4a + 8a2 = 7 Solving for variable 'a'. Reorder the terms: -24 + -7 + -4a + 8a2 = 7 + -7 Combine like terms: -24 + -7 = -31 -31 + -4a + 8a2 = 7 + -7 Combine like terms: 7 + -7 = 0 -31 + -4a + 8a2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -3.875 + -0.5a + a2 = 0 Move the constant term to the right: Add '3.875' to each side of the equation. -3.875 + -0.5a + 3.875 + a2 = 0 + 3.875 Reorder the terms: -3.875 + 3.875 + -0.5a + a2 = 0 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + -0.5a + a2 = 0 + 3.875 -0.5a + a2 = 0 + 3.875 Combine like terms: 0 + 3.875 = 3.875 -0.5a + a2 = 3.875 The a term is -0.5a. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5a + 0.0625 + a2 = 3.875 + 0.0625 Reorder the terms: 0.0625 + -0.5a + a2 = 3.875 + 0.0625 Combine like terms: 3.875 + 0.0625 = 3.9375 0.0625 + -0.5a + a2 = 3.9375 Factor a perfect square on the left side: (a + -0.25)(a + -0.25) = 3.9375 Calculate the square root of the right side: 1.984313483 Break this problem into two subproblems by setting (a + -0.25) equal to 1.984313483 and -1.984313483.Subproblem 1
a + -0.25 = 1.984313483 Simplifying a + -0.25 = 1.984313483 Reorder the terms: -0.25 + a = 1.984313483 Solving -0.25 + a = 1.984313483 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = 1.984313483 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 1.984313483 + 0.25 a = 1.984313483 + 0.25 Combine like terms: 1.984313483 + 0.25 = 2.234313483 a = 2.234313483 Simplifying a = 2.234313483Subproblem 2
a + -0.25 = -1.984313483 Simplifying a + -0.25 = -1.984313483 Reorder the terms: -0.25 + a = -1.984313483 Solving -0.25 + a = -1.984313483 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = -1.984313483 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -1.984313483 + 0.25 a = -1.984313483 + 0.25 Combine like terms: -1.984313483 + 0.25 = -1.734313483 a = -1.734313483 Simplifying a = -1.734313483Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.234313483, -1.734313483}
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