4x^2+5x-54=0

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Solution for 4x^2+5x-54=0 equation:


Simplifying
4x2 + 5x + -54 = 0

Reorder the terms:
-54 + 5x + 4x2 = 0

Solving
-54 + 5x + 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 + 1.25x + x2 = 0

Move the constant term to the right:

Add '13.5' to each side of the equation.
-13.5 + 1.25x + 13.5 + x2 = 0 + 13.5

Reorder the terms:
-13.5 + 13.5 + 1.25x + x2 = 0 + 13.5

Combine like terms: -13.5 + 13.5 = 0.0
0.0 + 1.25x + x2 = 0 + 13.5
1.25x + x2 = 0 + 13.5

Combine like terms: 0 + 13.5 = 13.5
1.25x + x2 = 13.5

The x term is 1.25x.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25x + 0.390625 + x2 = 13.5 + 0.390625

Reorder the terms:
0.390625 + 1.25x + x2 = 13.5 + 0.390625

Combine like terms: 13.5 + 0.390625 = 13.890625
0.390625 + 1.25x + x2 = 13.890625

Factor a perfect square on the left side:
(x + 0.625)(x + 0.625) = 13.890625

Calculate the square root of the right side: 3.727012879

Break this problem into two subproblems by setting 
(x + 0.625) equal to 3.727012879 and -3.727012879.

Subproblem 1

x + 0.625 = 3.727012879 Simplifying x + 0.625 = 3.727012879 Reorder the terms: 0.625 + x = 3.727012879 Solving 0.625 + x = 3.727012879 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = 3.727012879 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 3.727012879 + -0.625 x = 3.727012879 + -0.625 Combine like terms: 3.727012879 + -0.625 = 3.102012879 x = 3.102012879 Simplifying x = 3.102012879

Subproblem 2

x + 0.625 = -3.727012879 Simplifying x + 0.625 = -3.727012879 Reorder the terms: 0.625 + x = -3.727012879 Solving 0.625 + x = -3.727012879 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = -3.727012879 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -3.727012879 + -0.625 x = -3.727012879 + -0.625 Combine like terms: -3.727012879 + -0.625 = -4.352012879 x = -4.352012879 Simplifying x = -4.352012879

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.102012879, -4.352012879}

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