8x^2+51x+54=0

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Solution for 8x^2+51x+54=0 equation:


Simplifying
8x2 + 51x + 54 = 0

Reorder the terms:
54 + 51x + 8x2 = 0

Solving
54 + 51x + 8x2 = 0

Solving for variable 'x'.

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

Divide each side by '8'.
6.75 + 6.375x + x2 = 0

Move the constant term to the right:

Add '-6.75' to each side of the equation.
6.75 + 6.375x + -6.75 + x2 = 0 + -6.75

Reorder the terms:
6.75 + -6.75 + 6.375x + x2 = 0 + -6.75

Combine like terms: 6.75 + -6.75 = 0.00
0.00 + 6.375x + x2 = 0 + -6.75
6.375x + x2 = 0 + -6.75

Combine like terms: 0 + -6.75 = -6.75
6.375x + x2 = -6.75

The x term is 6.375x.  Take half its coefficient (3.1875).
Square it (10.16015625) and add it to both sides.

Add '10.16015625' to each side of the equation.
6.375x + 10.16015625 + x2 = -6.75 + 10.16015625

Reorder the terms:
10.16015625 + 6.375x + x2 = -6.75 + 10.16015625

Combine like terms: -6.75 + 10.16015625 = 3.41015625
10.16015625 + 6.375x + x2 = 3.41015625

Factor a perfect square on the left side:
(x + 3.1875)(x + 3.1875) = 3.41015625

Calculate the square root of the right side: 1.846660838

Break this problem into two subproblems by setting 
(x + 3.1875) equal to 1.846660838 and -1.846660838.

Subproblem 1

x + 3.1875 = 1.846660838 Simplifying x + 3.1875 = 1.846660838 Reorder the terms: 3.1875 + x = 1.846660838 Solving 3.1875 + x = 1.846660838 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.1875' to each side of the equation. 3.1875 + -3.1875 + x = 1.846660838 + -3.1875 Combine like terms: 3.1875 + -3.1875 = 0.0000 0.0000 + x = 1.846660838 + -3.1875 x = 1.846660838 + -3.1875 Combine like terms: 1.846660838 + -3.1875 = -1.340839162 x = -1.340839162 Simplifying x = -1.340839162

Subproblem 2

x + 3.1875 = -1.846660838 Simplifying x + 3.1875 = -1.846660838 Reorder the terms: 3.1875 + x = -1.846660838 Solving 3.1875 + x = -1.846660838 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.1875' to each side of the equation. 3.1875 + -3.1875 + x = -1.846660838 + -3.1875 Combine like terms: 3.1875 + -3.1875 = 0.0000 0.0000 + x = -1.846660838 + -3.1875 x = -1.846660838 + -3.1875 Combine like terms: -1.846660838 + -3.1875 = -5.034160838 x = -5.034160838 Simplifying x = -5.034160838

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.340839162, -5.034160838}

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