4x^2+9x-52=0

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Solution for 4x^2+9x-52=0 equation: 


Simplifying
4x2 + 9x + -52 = 0

Reorder the terms:
-52 + 9x + 4x2 = 0

Solving
-52 + 9x + 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 + 2.25x + x2 = 0

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + 2.25x + 13 + x2 = 0 + 13

Reorder the terms:
-13 + 13 + 2.25x + x2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + 2.25x + x2 = 0 + 13
2.25x + x2 = 0 + 13

Combine like terms: 0 + 13 = 13
2.25x + x2 = 13

The x term is 2.25x.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25x + 1.265625 + x2 = 13 + 1.265625

Reorder the terms:
1.265625 + 2.25x + x2 = 13 + 1.265625

Combine like terms: 13 + 1.265625 = 14.265625
1.265625 + 2.25x + x2 = 14.265625

Factor a perfect square on the left side:
(x + 1.125)(x + 1.125) = 14.265625

Calculate the square root of the right side: 3.776986232

Break this problem into two subproblems by setting 
(x + 1.125) equal to 3.776986232 and -3.776986232.


Subproblem 1
x + 1.125 = 3.776986232

Simplifying
x + 1.125 = 3.776986232

Reorder the terms:
1.125 + x = 3.776986232

Solving
1.125 + x = 3.776986232

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.125' to each side of the equation.
1.125 + -1.125 + x = 3.776986232 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + x = 3.776986232 + -1.125
x = 3.776986232 + -1.125

Combine like terms: 3.776986232 + -1.125 = 2.651986232
x = 2.651986232

Simplifying
x = 2.651986232


Subproblem 2
x + 1.125 = -3.776986232

Simplifying
x + 1.125 = -3.776986232

Reorder the terms:
1.125 + x = -3.776986232

Solving
1.125 + x = -3.776986232

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.125' to each side of the equation.
1.125 + -1.125 + x = -3.776986232 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + x = -3.776986232 + -1.125
x = -3.776986232 + -1.125

Combine like terms: -3.776986232 + -1.125 = -4.901986232
x = -4.901986232

Simplifying
x = -4.901986232


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.651986232, -4.901986232}

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