3x^2+4x-11.25=0

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Solution for 3x^2+4x-11.25=0 equation:


Simplifying
3x2 + 4x + -11.25 = 0

Reorder the terms:
-11.25 + 4x + 3x2 = 0

Solving
-11.25 + 4x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-3.75 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '3.75' to each side of the equation.
-3.75 + 1.333333333x + 3.75 + x2 = 0 + 3.75

Reorder the terms:
-3.75 + 3.75 + 1.333333333x + x2 = 0 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + 1.333333333x + x2 = 0 + 3.75
1.333333333x + x2 = 0 + 3.75

Combine like terms: 0 + 3.75 = 3.75
1.333333333x + x2 = 3.75

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 3.75 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 3.75 + 0.4444444442

Combine like terms: 3.75 + 0.4444444442 = 4.1944444442
0.4444444442 + 1.333333333x + x2 = 4.1944444442

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 4.1944444442

Calculate the square root of the right side: 2.048034288

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 2.048034288 and -2.048034288.

Subproblem 1

x + 0.6666666665 = 2.048034288 Simplifying x + 0.6666666665 = 2.048034288 Reorder the terms: 0.6666666665 + x = 2.048034288 Solving 0.6666666665 + x = 2.048034288 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 2.048034288 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 2.048034288 + -0.6666666665 x = 2.048034288 + -0.6666666665 Combine like terms: 2.048034288 + -0.6666666665 = 1.3813676215 x = 1.3813676215 Simplifying x = 1.3813676215

Subproblem 2

x + 0.6666666665 = -2.048034288 Simplifying x + 0.6666666665 = -2.048034288 Reorder the terms: 0.6666666665 + x = -2.048034288 Solving 0.6666666665 + x = -2.048034288 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -2.048034288 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -2.048034288 + -0.6666666665 x = -2.048034288 + -0.6666666665 Combine like terms: -2.048034288 + -0.6666666665 = -2.7147009545 x = -2.7147009545 Simplifying x = -2.7147009545

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3813676215, -2.7147009545}

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