4a^2-6a-27=0

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Solution for 4a^2-6a-27=0 equation:


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 = 3.454163457

Subproblem 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 = -1.954163457

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.454163457, -1.954163457}

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