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Simplifying (4a2 + 3a + -7) + (-2a2 + a + -15) = 0 Reorder the terms: (-7 + 3a + 4a2) + (-2a2 + a + -15) = 0 Remove parenthesis around (-7 + 3a + 4a2) -7 + 3a + 4a2 + (-2a2 + a + -15) = 0 Reorder the terms: -7 + 3a + 4a2 + (-15 + a + -2a2) = 0 Remove parenthesis around (-15 + a + -2a2) -7 + 3a + 4a2 + -15 + a + -2a2 = 0 Reorder the terms: -7 + -15 + 3a + a + 4a2 + -2a2 = 0 Combine like terms: -7 + -15 = -22 -22 + 3a + a + 4a2 + -2a2 = 0 Combine like terms: 3a + a = 4a -22 + 4a + 4a2 + -2a2 = 0 Combine like terms: 4a2 + -2a2 = 2a2 -22 + 4a + 2a2 = 0 Solving -22 + 4a + 2a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '2'. 2(-11 + 2a + a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-11 + 2a + a2)' equal to zero and attempt to solve: Simplifying -11 + 2a + a2 = 0 Solving -11 + 2a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '11' to each side of the equation. -11 + 2a + 11 + a2 = 0 + 11 Reorder the terms: -11 + 11 + 2a + a2 = 0 + 11 Combine like terms: -11 + 11 = 0 0 + 2a + a2 = 0 + 11 2a + a2 = 0 + 11 Combine like terms: 0 + 11 = 11 2a + a2 = 11 The a term is 2a. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2a + 1 + a2 = 11 + 1 Reorder the terms: 1 + 2a + a2 = 11 + 1 Combine like terms: 11 + 1 = 12 1 + 2a + a2 = 12 Factor a perfect square on the left side: (a + 1)(a + 1) = 12 Calculate the square root of the right side: 3.464101615 Break this problem into two subproblems by setting (a + 1) equal to 3.464101615 and -3.464101615.Subproblem 1
a + 1 = 3.464101615 Simplifying a + 1 = 3.464101615 Reorder the terms: 1 + a = 3.464101615 Solving 1 + a = 3.464101615 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = 3.464101615 + -1 Combine like terms: 1 + -1 = 0 0 + a = 3.464101615 + -1 a = 3.464101615 + -1 Combine like terms: 3.464101615 + -1 = 2.464101615 a = 2.464101615 Simplifying a = 2.464101615Subproblem 2
a + 1 = -3.464101615 Simplifying a + 1 = -3.464101615 Reorder the terms: 1 + a = -3.464101615 Solving 1 + a = -3.464101615 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = -3.464101615 + -1 Combine like terms: 1 + -1 = 0 0 + a = -3.464101615 + -1 a = -3.464101615 + -1 Combine like terms: -3.464101615 + -1 = -4.464101615 a = -4.464101615 Simplifying a = -4.464101615Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.464101615, -4.464101615}Solution
a = {2.464101615, -4.464101615}
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