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Simplifying 2(a + -7) * a + 15 = 0 Reorder the terms: 2(-7 + a) * a + 15 = 0 Reorder the terms for easier multiplication: 2a(-7 + a) + 15 = 0 (-7 * 2a + a * 2a) + 15 = 0 (-14a + 2a2) + 15 = 0 Reorder the terms: 15 + -14a + 2a2 = 0 Solving 15 + -14a + 2a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 7.5 + -7a + a2 = 0 Move the constant term to the right: Add '-7.5' to each side of the equation. 7.5 + -7a + -7.5 + a2 = 0 + -7.5 Reorder the terms: 7.5 + -7.5 + -7a + a2 = 0 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + -7a + a2 = 0 + -7.5 -7a + a2 = 0 + -7.5 Combine like terms: 0 + -7.5 = -7.5 -7a + a2 = -7.5 The a term is -7a. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7a + 12.25 + a2 = -7.5 + 12.25 Reorder the terms: 12.25 + -7a + a2 = -7.5 + 12.25 Combine like terms: -7.5 + 12.25 = 4.75 12.25 + -7a + a2 = 4.75 Factor a perfect square on the left side: (a + -3.5)(a + -3.5) = 4.75 Calculate the square root of the right side: 2.179449472 Break this problem into two subproblems by setting (a + -3.5) equal to 2.179449472 and -2.179449472.Subproblem 1
a + -3.5 = 2.179449472 Simplifying a + -3.5 = 2.179449472 Reorder the terms: -3.5 + a = 2.179449472 Solving -3.5 + a = 2.179449472 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + a = 2.179449472 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + a = 2.179449472 + 3.5 a = 2.179449472 + 3.5 Combine like terms: 2.179449472 + 3.5 = 5.679449472 a = 5.679449472 Simplifying a = 5.679449472Subproblem 2
a + -3.5 = -2.179449472 Simplifying a + -3.5 = -2.179449472 Reorder the terms: -3.5 + a = -2.179449472 Solving -3.5 + a = -2.179449472 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + a = -2.179449472 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + a = -2.179449472 + 3.5 a = -2.179449472 + 3.5 Combine like terms: -2.179449472 + 3.5 = 1.320550528 a = 1.320550528 Simplifying a = 1.320550528Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.679449472, 1.320550528}
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