6a^2+51a+10=0

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Solution for 6a^2+51a+10=0 equation:


Simplifying
6a2 + 51a + 10 = 0

Reorder the terms:
10 + 51a + 6a2 = 0

Solving
10 + 51a + 6a2 = 0

Solving for variable 'a'.

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
1.666666667 + 8.5a + a2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + 8.5a + -1.666666667 + a2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + 8.5a + a2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + 8.5a + a2 = 0 + -1.666666667
8.5a + a2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
8.5a + a2 = -1.666666667

The a term is 8.5a.  Take half its coefficient (4.25).
Square it (18.0625) and add it to both sides.

Add '18.0625' to each side of the equation.
8.5a + 18.0625 + a2 = -1.666666667 + 18.0625

Reorder the terms:
18.0625 + 8.5a + a2 = -1.666666667 + 18.0625

Combine like terms: -1.666666667 + 18.0625 = 16.395833333
18.0625 + 8.5a + a2 = 16.395833333

Factor a perfect square on the left side:
(a + 4.25)(a + 4.25) = 16.395833333

Calculate the square root of the right side: 4.049176871

Break this problem into two subproblems by setting 
(a + 4.25) equal to 4.049176871 and -4.049176871.

Subproblem 1

a + 4.25 = 4.049176871 Simplifying a + 4.25 = 4.049176871 Reorder the terms: 4.25 + a = 4.049176871 Solving 4.25 + a = 4.049176871 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + a = 4.049176871 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + a = 4.049176871 + -4.25 a = 4.049176871 + -4.25 Combine like terms: 4.049176871 + -4.25 = -0.200823129 a = -0.200823129 Simplifying a = -0.200823129

Subproblem 2

a + 4.25 = -4.049176871 Simplifying a + 4.25 = -4.049176871 Reorder the terms: 4.25 + a = -4.049176871 Solving 4.25 + a = -4.049176871 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + a = -4.049176871 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + a = -4.049176871 + -4.25 a = -4.049176871 + -4.25 Combine like terms: -4.049176871 + -4.25 = -8.299176871 a = -8.299176871 Simplifying a = -8.299176871

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.200823129, -8.299176871}

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