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Simplifying 4a2 + -6a + -27 = 0 Reorder the terms: -27 + -6a + 4a2 = 0 Solving -27 + -6a + 4a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -6.75 + -1.5a + a2 = 0 Move the constant term to the right: Add '6.75' to each side of the equation. -6.75 + -1.5a + 6.75 + a2 = 0 + 6.75 Reorder the terms: -6.75 + 6.75 + -1.5a + a2 = 0 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + -1.5a + a2 = 0 + 6.75 -1.5a + a2 = 0 + 6.75 Combine like terms: 0 + 6.75 = 6.75 -1.5a + a2 = 6.75 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 = 6.75 + 0.5625 Reorder the terms: 0.5625 + -1.5a + a2 = 6.75 + 0.5625 Combine like terms: 6.75 + 0.5625 = 7.3125 0.5625 + -1.5a + a2 = 7.3125 Factor a perfect square on the left side: (a + -0.75)(a + -0.75) = 7.3125 Calculate the square root of the right side: 2.704163457 Break this problem into two subproblems by setting (a + -0.75) equal to 2.704163457 and -2.704163457.Subproblem 1
a + -0.75 = 2.704163457 Simplifying a + -0.75 = 2.704163457 Reorder the terms: -0.75 + a = 2.704163457 Solving -0.75 + a = 2.704163457 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 = 2.704163457 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + a = 2.704163457 + 0.75 a = 2.704163457 + 0.75 Combine like terms: 2.704163457 + 0.75 = 3.454163457 a = 3.454163457 Simplifying a = 3.454163457Subproblem 2
a + -0.75 = -2.704163457 Simplifying a + -0.75 = -2.704163457 Reorder the terms: -0.75 + a = -2.704163457 Solving -0.75 + a = -2.704163457 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 = -2.704163457 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + a = -2.704163457 + 0.75 a = -2.704163457 + 0.75 Combine like terms: -2.704163457 + 0.75 = -1.954163457 a = -1.954163457 Simplifying a = -1.954163457Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.454163457, -1.954163457}
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