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Simplifying 2a2 + 3a + -7 = 15 Reorder the terms: -7 + 3a + 2a2 = 15 Solving -7 + 3a + 2a2 = 15 Solving for variable 'a'. Reorder the terms: -7 + -15 + 3a + 2a2 = 15 + -15 Combine like terms: -7 + -15 = -22 -22 + 3a + 2a2 = 15 + -15 Combine like terms: 15 + -15 = 0 -22 + 3a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -11 + 1.5a + a2 = 0 Move the constant term to the right: Add '11' to each side of the equation. -11 + 1.5a + 11 + a2 = 0 + 11 Reorder the terms: -11 + 11 + 1.5a + a2 = 0 + 11 Combine like terms: -11 + 11 = 0 0 + 1.5a + a2 = 0 + 11 1.5a + a2 = 0 + 11 Combine like terms: 0 + 11 = 11 1.5a + a2 = 11 The a term is 1.5a. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5a + 0.5625 + a2 = 11 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 11 + 0.5625 Combine like terms: 11 + 0.5625 = 11.5625 0.5625 + 1.5a + a2 = 11.5625 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 11.5625 Calculate the square root of the right side: 3.400367627 Break this problem into two subproblems by setting (a + 0.75) equal to 3.400367627 and -3.400367627.Subproblem 1
a + 0.75 = 3.400367627 Simplifying a + 0.75 = 3.400367627 Reorder the terms: 0.75 + a = 3.400367627 Solving 0.75 + a = 3.400367627 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = 3.400367627 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 3.400367627 + -0.75 a = 3.400367627 + -0.75 Combine like terms: 3.400367627 + -0.75 = 2.650367627 a = 2.650367627 Simplifying a = 2.650367627Subproblem 2
a + 0.75 = -3.400367627 Simplifying a + 0.75 = -3.400367627 Reorder the terms: 0.75 + a = -3.400367627 Solving 0.75 + a = -3.400367627 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = -3.400367627 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -3.400367627 + -0.75 a = -3.400367627 + -0.75 Combine like terms: -3.400367627 + -0.75 = -4.150367627 a = -4.150367627 Simplifying a = -4.150367627Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.650367627, -4.150367627}
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