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Simplifying (13a2 + 13a + -8) + -1(-4a2 + 3a + 4) = 0 Reorder the terms: (-8 + 13a + 13a2) + -1(-4a2 + 3a + 4) = 0 Remove parenthesis around (-8 + 13a + 13a2) -8 + 13a + 13a2 + -1(-4a2 + 3a + 4) = 0 Reorder the terms: -8 + 13a + 13a2 + -1(4 + 3a + -4a2) = 0 -8 + 13a + 13a2 + (4 * -1 + 3a * -1 + -4a2 * -1) = 0 -8 + 13a + 13a2 + (-4 + -3a + 4a2) = 0 Reorder the terms: -8 + -4 + 13a + -3a + 13a2 + 4a2 = 0 Combine like terms: -8 + -4 = -12 -12 + 13a + -3a + 13a2 + 4a2 = 0 Combine like terms: 13a + -3a = 10a -12 + 10a + 13a2 + 4a2 = 0 Combine like terms: 13a2 + 4a2 = 17a2 -12 + 10a + 17a2 = 0 Solving -12 + 10a + 17a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 17 the coefficient of the squared term: Divide each side by '17'. -0.7058823529 + 0.5882352941a + a2 = 0 Move the constant term to the right: Add '0.7058823529' to each side of the equation. -0.7058823529 + 0.5882352941a + 0.7058823529 + a2 = 0 + 0.7058823529 Reorder the terms: -0.7058823529 + 0.7058823529 + 0.5882352941a + a2 = 0 + 0.7058823529 Combine like terms: -0.7058823529 + 0.7058823529 = 0.0000000000 0.0000000000 + 0.5882352941a + a2 = 0 + 0.7058823529 0.5882352941a + a2 = 0 + 0.7058823529 Combine like terms: 0 + 0.7058823529 = 0.7058823529 0.5882352941a + a2 = 0.7058823529 The a term is 0.5882352941a. Take half its coefficient (0.2941176471). Square it (0.08650519034) and add it to both sides. Add '0.08650519034' to each side of the equation. 0.5882352941a + 0.08650519034 + a2 = 0.7058823529 + 0.08650519034 Reorder the terms: 0.08650519034 + 0.5882352941a + a2 = 0.7058823529 + 0.08650519034 Combine like terms: 0.7058823529 + 0.08650519034 = 0.79238754324 0.08650519034 + 0.5882352941a + a2 = 0.79238754324 Factor a perfect square on the left side: (a + 0.2941176471)(a + 0.2941176471) = 0.79238754324 Calculate the square root of the right side: 0.890161526 Break this problem into two subproblems by setting (a + 0.2941176471) equal to 0.890161526 and -0.890161526.Subproblem 1
a + 0.2941176471 = 0.890161526 Simplifying a + 0.2941176471 = 0.890161526 Reorder the terms: 0.2941176471 + a = 0.890161526 Solving 0.2941176471 + a = 0.890161526 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.2941176471' to each side of the equation. 0.2941176471 + -0.2941176471 + a = 0.890161526 + -0.2941176471 Combine like terms: 0.2941176471 + -0.2941176471 = 0.0000000000 0.0000000000 + a = 0.890161526 + -0.2941176471 a = 0.890161526 + -0.2941176471 Combine like terms: 0.890161526 + -0.2941176471 = 0.5960438789 a = 0.5960438789 Simplifying a = 0.5960438789Subproblem 2
a + 0.2941176471 = -0.890161526 Simplifying a + 0.2941176471 = -0.890161526 Reorder the terms: 0.2941176471 + a = -0.890161526 Solving 0.2941176471 + a = -0.890161526 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.2941176471' to each side of the equation. 0.2941176471 + -0.2941176471 + a = -0.890161526 + -0.2941176471 Combine like terms: 0.2941176471 + -0.2941176471 = 0.0000000000 0.0000000000 + a = -0.890161526 + -0.2941176471 a = -0.890161526 + -0.2941176471 Combine like terms: -0.890161526 + -0.2941176471 = -1.1842791731 a = -1.1842791731 Simplifying a = -1.1842791731Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.5960438789, -1.1842791731}
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