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Simplifying 2a2 + 3a + -7 = 55 Reorder the terms: -7 + 3a + 2a2 = 55 Solving -7 + 3a + 2a2 = 55 Solving for variable 'a'. Reorder the terms: -7 + -55 + 3a + 2a2 = 55 + -55 Combine like terms: -7 + -55 = -62 -62 + 3a + 2a2 = 55 + -55 Combine like terms: 55 + -55 = 0 -62 + 3a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -31 + 1.5a + a2 = 0 Move the constant term to the right: Add '31' to each side of the equation. -31 + 1.5a + 31 + a2 = 0 + 31 Reorder the terms: -31 + 31 + 1.5a + a2 = 0 + 31 Combine like terms: -31 + 31 = 0 0 + 1.5a + a2 = 0 + 31 1.5a + a2 = 0 + 31 Combine like terms: 0 + 31 = 31 1.5a + a2 = 31 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 = 31 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 31 + 0.5625 Combine like terms: 31 + 0.5625 = 31.5625 0.5625 + 1.5a + a2 = 31.5625 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 31.5625 Calculate the square root of the right side: 5.618051264 Break this problem into two subproblems by setting (a + 0.75) equal to 5.618051264 and -5.618051264.Subproblem 1
a + 0.75 = 5.618051264 Simplifying a + 0.75 = 5.618051264 Reorder the terms: 0.75 + a = 5.618051264 Solving 0.75 + a = 5.618051264 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 = 5.618051264 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 5.618051264 + -0.75 a = 5.618051264 + -0.75 Combine like terms: 5.618051264 + -0.75 = 4.868051264 a = 4.868051264 Simplifying a = 4.868051264Subproblem 2
a + 0.75 = -5.618051264 Simplifying a + 0.75 = -5.618051264 Reorder the terms: 0.75 + a = -5.618051264 Solving 0.75 + a = -5.618051264 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 = -5.618051264 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -5.618051264 + -0.75 a = -5.618051264 + -0.75 Combine like terms: -5.618051264 + -0.75 = -6.368051264 a = -6.368051264 Simplifying a = -6.368051264Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.868051264, -6.368051264}
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