a^2+20a-65=6

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Solution for a^2+20a-65=6 equation:


Simplifying
a2 + 20a + -65 = 6

Reorder the terms:
-65 + 20a + a2 = 6

Solving
-65 + 20a + a2 = 6

Solving for variable 'a'.

Reorder the terms:
-65 + -6 + 20a + a2 = 6 + -6

Combine like terms: -65 + -6 = -71
-71 + 20a + a2 = 6 + -6

Combine like terms: 6 + -6 = 0
-71 + 20a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '71' to each side of the equation.
-71 + 20a + 71 + a2 = 0 + 71

Reorder the terms:
-71 + 71 + 20a + a2 = 0 + 71

Combine like terms: -71 + 71 = 0
0 + 20a + a2 = 0 + 71
20a + a2 = 0 + 71

Combine like terms: 0 + 71 = 71
20a + a2 = 71

The a term is 20a.  Take half its coefficient (10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
20a + 100 + a2 = 71 + 100

Reorder the terms:
100 + 20a + a2 = 71 + 100

Combine like terms: 71 + 100 = 171
100 + 20a + a2 = 171

Factor a perfect square on the left side:
(a + 10)(a + 10) = 171

Calculate the square root of the right side: 13.076696831

Break this problem into two subproblems by setting 
(a + 10) equal to 13.076696831 and -13.076696831.

Subproblem 1

a + 10 = 13.076696831 Simplifying a + 10 = 13.076696831 Reorder the terms: 10 + a = 13.076696831 Solving 10 + a = 13.076696831 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + a = 13.076696831 + -10 Combine like terms: 10 + -10 = 0 0 + a = 13.076696831 + -10 a = 13.076696831 + -10 Combine like terms: 13.076696831 + -10 = 3.076696831 a = 3.076696831 Simplifying a = 3.076696831

Subproblem 2

a + 10 = -13.076696831 Simplifying a + 10 = -13.076696831 Reorder the terms: 10 + a = -13.076696831 Solving 10 + a = -13.076696831 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + a = -13.076696831 + -10 Combine like terms: 10 + -10 = 0 0 + a = -13.076696831 + -10 a = -13.076696831 + -10 Combine like terms: -13.076696831 + -10 = -23.076696831 a = -23.076696831 Simplifying a = -23.076696831

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.076696831, -23.076696831}

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