4x^2-69x+27=0

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Solution for 4x^2-69x+27=0 equation:


Simplifying
4x2 + -69x + 27 = 0

Reorder the terms:
27 + -69x + 4x2 = 0

Solving
27 + -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'.
6.75 + -17.25x + x2 = 0

Move the constant term to the right:

Add '-6.75' to each side of the equation.
6.75 + -17.25x + -6.75 + x2 = 0 + -6.75

Reorder the terms:
6.75 + -6.75 + -17.25x + x2 = 0 + -6.75

Combine like terms: 6.75 + -6.75 = 0.00
0.00 + -17.25x + x2 = 0 + -6.75
-17.25x + x2 = 0 + -6.75

Combine like terms: 0 + -6.75 = -6.75
-17.25x + x2 = -6.75

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 = -6.75 + 74.390625

Reorder the terms:
74.390625 + -17.25x + x2 = -6.75 + 74.390625

Combine like terms: -6.75 + 74.390625 = 67.640625
74.390625 + -17.25x + x2 = 67.640625

Factor a perfect square on the left side:
(x + -8.625)(x + -8.625) = 67.640625

Calculate the square root of the right side: 8.224392075

Break this problem into two subproblems by setting 
(x + -8.625) equal to 8.224392075 and -8.224392075.

Subproblem 1

x + -8.625 = 8.224392075 Simplifying x + -8.625 = 8.224392075 Reorder the terms: -8.625 + x = 8.224392075 Solving -8.625 + x = 8.224392075 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 = 8.224392075 + 8.625 Combine like terms: -8.625 + 8.625 = 0.000 0.000 + x = 8.224392075 + 8.625 x = 8.224392075 + 8.625 Combine like terms: 8.224392075 + 8.625 = 16.849392075 x = 16.849392075 Simplifying x = 16.849392075

Subproblem 2

x + -8.625 = -8.224392075 Simplifying x + -8.625 = -8.224392075 Reorder the terms: -8.625 + x = -8.224392075 Solving -8.625 + x = -8.224392075 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 = -8.224392075 + 8.625 Combine like terms: -8.625 + 8.625 = 0.000 0.000 + x = -8.224392075 + 8.625 x = -8.224392075 + 8.625 Combine like terms: -8.224392075 + 8.625 = 0.400607925 x = 0.400607925 Simplifying x = 0.400607925

Solution

The solution to the problem is based on the solutions from the subproblems. x = {16.849392075, 0.400607925}

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