a^2-3a-11=0

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Solution for a^2-3a-11=0 equation:


Simplifying
a2 + -3a + -11 = 0

Reorder the terms:
-11 + -3a + a2 = 0

Solving
-11 + -3a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + -3a + 11 + a2 = 0 + 11

Reorder the terms:
-11 + 11 + -3a + a2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + -3a + a2 = 0 + 11
-3a + a2 = 0 + 11

Combine like terms: 0 + 11 = 11
-3a + a2 = 11

The a term is -3a.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3a + 2.25 + a2 = 11 + 2.25

Reorder the terms:
2.25 + -3a + a2 = 11 + 2.25

Combine like terms: 11 + 2.25 = 13.25
2.25 + -3a + a2 = 13.25

Factor a perfect square on the left side:
(a + -1.5)(a + -1.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(a + -1.5) equal to 3.640054945 and -3.640054945.

Subproblem 1

a + -1.5 = 3.640054945 Simplifying a + -1.5 = 3.640054945 Reorder the terms: -1.5 + a = 3.640054945 Solving -1.5 + a = 3.640054945 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = 3.640054945 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = 3.640054945 + 1.5 a = 3.640054945 + 1.5 Combine like terms: 3.640054945 + 1.5 = 5.140054945 a = 5.140054945 Simplifying a = 5.140054945

Subproblem 2

a + -1.5 = -3.640054945 Simplifying a + -1.5 = -3.640054945 Reorder the terms: -1.5 + a = -3.640054945 Solving -1.5 + a = -3.640054945 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = -3.640054945 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = -3.640054945 + 1.5 a = -3.640054945 + 1.5 Combine like terms: -3.640054945 + 1.5 = -2.140054945 a = -2.140054945 Simplifying a = -2.140054945

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.140054945, -2.140054945}

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