9x(x+6)-(3x+1)=1

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Solution for 9x(x+6)-(3x+1)=1 equation:


Simplifying
9x(x + 6) + -1(3x + 1) = 1

Reorder the terms:
9x(6 + x) + -1(3x + 1) = 1
(6 * 9x + x * 9x) + -1(3x + 1) = 1
(54x + 9x2) + -1(3x + 1) = 1

Reorder the terms:
54x + 9x2 + -1(1 + 3x) = 1
54x + 9x2 + (1 * -1 + 3x * -1) = 1
54x + 9x2 + (-1 + -3x) = 1

Reorder the terms:
-1 + 54x + -3x + 9x2 = 1

Combine like terms: 54x + -3x = 51x
-1 + 51x + 9x2 = 1

Solving
-1 + 51x + 9x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + -1 + 51x + 9x2 = 1 + -1

Combine like terms: -1 + -1 = -2
-2 + 51x + 9x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-2 + 51x + 9x2 = 0

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

Divide each side by '9'.
-0.2222222222 + 5.666666667x + x2 = 0

Move the constant term to the right:

Add '0.2222222222' to each side of the equation.
-0.2222222222 + 5.666666667x + 0.2222222222 + x2 = 0 + 0.2222222222

Reorder the terms:
-0.2222222222 + 0.2222222222 + 5.666666667x + x2 = 0 + 0.2222222222

Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000
0.0000000000 + 5.666666667x + x2 = 0 + 0.2222222222
5.666666667x + x2 = 0 + 0.2222222222

Combine like terms: 0 + 0.2222222222 = 0.2222222222
5.666666667x + x2 = 0.2222222222

The x term is 5.666666667x.  Take half its coefficient (2.833333334).
Square it (8.027777782) and add it to both sides.

Add '8.027777782' to each side of the equation.
5.666666667x + 8.027777782 + x2 = 0.2222222222 + 8.027777782

Reorder the terms:
8.027777782 + 5.666666667x + x2 = 0.2222222222 + 8.027777782

Combine like terms: 0.2222222222 + 8.027777782 = 8.2500000042
8.027777782 + 5.666666667x + x2 = 8.2500000042

Factor a perfect square on the left side:
(x + 2.833333334)(x + 2.833333334) = 8.2500000042

Calculate the square root of the right side: 2.872281324

Break this problem into two subproblems by setting 
(x + 2.833333334) equal to 2.872281324 and -2.872281324.

Subproblem 1

x + 2.833333334 = 2.872281324 Simplifying x + 2.833333334 = 2.872281324 Reorder the terms: 2.833333334 + x = 2.872281324 Solving 2.833333334 + x = 2.872281324 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.833333334' to each side of the equation. 2.833333334 + -2.833333334 + x = 2.872281324 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + x = 2.872281324 + -2.833333334 x = 2.872281324 + -2.833333334 Combine like terms: 2.872281324 + -2.833333334 = 0.03894799 x = 0.03894799 Simplifying x = 0.03894799

Subproblem 2

x + 2.833333334 = -2.872281324 Simplifying x + 2.833333334 = -2.872281324 Reorder the terms: 2.833333334 + x = -2.872281324 Solving 2.833333334 + x = -2.872281324 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.833333334' to each side of the equation. 2.833333334 + -2.833333334 + x = -2.872281324 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + x = -2.872281324 + -2.833333334 x = -2.872281324 + -2.833333334 Combine like terms: -2.872281324 + -2.833333334 = -5.705614658 x = -5.705614658 Simplifying x = -5.705614658

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.03894799, -5.705614658}

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