4x^2-27x-53=0

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Solution for 4x^2-27x-53=0 equation:


Simplifying
4x2 + -27x + -53 = 0

Reorder the terms:
-53 + -27x + 4x2 = 0

Solving
-53 + -27x + 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'.
-13.25 + -6.75x + x2 = 0

Move the constant term to the right:

Add '13.25' to each side of the equation.
-13.25 + -6.75x + 13.25 + x2 = 0 + 13.25

Reorder the terms:
-13.25 + 13.25 + -6.75x + x2 = 0 + 13.25

Combine like terms: -13.25 + 13.25 = 0.00
0.00 + -6.75x + x2 = 0 + 13.25
-6.75x + x2 = 0 + 13.25

Combine like terms: 0 + 13.25 = 13.25
-6.75x + x2 = 13.25

The x term is -6.75x.  Take half its coefficient (-3.375).
Square it (11.390625) and add it to both sides.

Add '11.390625' to each side of the equation.
-6.75x + 11.390625 + x2 = 13.25 + 11.390625

Reorder the terms:
11.390625 + -6.75x + x2 = 13.25 + 11.390625

Combine like terms: 13.25 + 11.390625 = 24.640625
11.390625 + -6.75x + x2 = 24.640625

Factor a perfect square on the left side:
(x + -3.375)(x + -3.375) = 24.640625

Calculate the square root of the right side: 4.963932413

Break this problem into two subproblems by setting 
(x + -3.375) equal to 4.963932413 and -4.963932413.

Subproblem 1

x + -3.375 = 4.963932413 Simplifying x + -3.375 = 4.963932413 Reorder the terms: -3.375 + x = 4.963932413 Solving -3.375 + x = 4.963932413 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.375' to each side of the equation. -3.375 + 3.375 + x = 4.963932413 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + x = 4.963932413 + 3.375 x = 4.963932413 + 3.375 Combine like terms: 4.963932413 + 3.375 = 8.338932413 x = 8.338932413 Simplifying x = 8.338932413

Subproblem 2

x + -3.375 = -4.963932413 Simplifying x + -3.375 = -4.963932413 Reorder the terms: -3.375 + x = -4.963932413 Solving -3.375 + x = -4.963932413 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.375' to each side of the equation. -3.375 + 3.375 + x = -4.963932413 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + x = -4.963932413 + 3.375 x = -4.963932413 + 3.375 Combine like terms: -4.963932413 + 3.375 = -1.588932413 x = -1.588932413 Simplifying x = -1.588932413

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.338932413, -1.588932413}

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