4x^2-69x+54=0

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Solution for 4x^2-69x+54=0 equation:


Simplifying
4x2 + -69x + 54 = 0

Reorder the terms:
54 + -69x + 4x2 = 0

Solving
54 + -69x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
13.5 + -17.25x + x2 = 0

Move the constant term to the right:

Add '-13.5' to each side of the equation.
13.5 + -17.25x + -13.5 + x2 = 0 + -13.5

Reorder the terms:
13.5 + -13.5 + -17.25x + x2 = 0 + -13.5

Combine like terms: 13.5 + -13.5 = 0.0
0.0 + -17.25x + x2 = 0 + -13.5
-17.25x + x2 = 0 + -13.5

Combine like terms: 0 + -13.5 = -13.5
-17.25x + x2 = -13.5

The x term is -17.25x.  Take half its coefficient (-8.625).
Square it (74.390625) and add it to both sides.

Add '74.390625' to each side of the equation.
-17.25x + 74.390625 + x2 = -13.5 + 74.390625

Reorder the terms:
74.390625 + -17.25x + x2 = -13.5 + 74.390625

Combine like terms: -13.5 + 74.390625 = 60.890625
74.390625 + -17.25x + x2 = 60.890625

Factor a perfect square on the left side:
(x + -8.625)(x + -8.625) = 60.890625

Calculate the square root of the right side: 7.803244518

Break this problem into two subproblems by setting 
(x + -8.625) equal to 7.803244518 and -7.803244518.

Subproblem 1

x + -8.625 = 7.803244518 Simplifying x + -8.625 = 7.803244518 Reorder the terms: -8.625 + x = 7.803244518 Solving -8.625 + x = 7.803244518 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.625' to each side of the equation. -8.625 + 8.625 + x = 7.803244518 + 8.625 Combine like terms: -8.625 + 8.625 = 0.000 0.000 + x = 7.803244518 + 8.625 x = 7.803244518 + 8.625 Combine like terms: 7.803244518 + 8.625 = 16.428244518 x = 16.428244518 Simplifying x = 16.428244518

Subproblem 2

x + -8.625 = -7.803244518 Simplifying x + -8.625 = -7.803244518 Reorder the terms: -8.625 + x = -7.803244518 Solving -8.625 + x = -7.803244518 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.625' to each side of the equation. -8.625 + 8.625 + x = -7.803244518 + 8.625 Combine like terms: -8.625 + 8.625 = 0.000 0.000 + x = -7.803244518 + 8.625 x = -7.803244518 + 8.625 Combine like terms: -7.803244518 + 8.625 = 0.821755482 x = 0.821755482 Simplifying x = 0.821755482

Solution

The solution to the problem is based on the solutions from the subproblems. x = {16.428244518, 0.821755482}

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