4x^2+x-45=0

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Solution for 4x^2+x-45=0 equation:


Simplifying
4x2 + x + -45 = 0

Reorder the terms:
-45 + x + 4x2 = 0

Solving
-45 + x + 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'.
-11.25 + 0.25x + x2 = 0

Move the constant term to the right:

Add '11.25' to each side of the equation.
-11.25 + 0.25x + 11.25 + x2 = 0 + 11.25

Reorder the terms:
-11.25 + 11.25 + 0.25x + x2 = 0 + 11.25

Combine like terms: -11.25 + 11.25 = 0.00
0.00 + 0.25x + x2 = 0 + 11.25
0.25x + x2 = 0 + 11.25

Combine like terms: 0 + 11.25 = 11.25
0.25x + x2 = 11.25

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.25x + 0.25 + x2 = 11.25 + 0.25

Reorder the terms:
0.25 + 0.25x + x2 = 11.25 + 0.25

Combine like terms: 11.25 + 0.25 = 11.5
0.25 + 0.25x + x2 = 11.5

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 11.5

Calculate the square root of the right side: 3.391164992

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.391164992 and -3.391164992.

Subproblem 1

x + 0.5 = 3.391164992 Simplifying x + 0.5 = 3.391164992 Reorder the terms: 0.5 + x = 3.391164992 Solving 0.5 + x = 3.391164992 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 3.391164992 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.391164992 + -0.5 x = 3.391164992 + -0.5 Combine like terms: 3.391164992 + -0.5 = 2.891164992 x = 2.891164992 Simplifying x = 2.891164992

Subproblem 2

x + 0.5 = -3.391164992 Simplifying x + 0.5 = -3.391164992 Reorder the terms: 0.5 + x = -3.391164992 Solving 0.5 + x = -3.391164992 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -3.391164992 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.391164992 + -0.5 x = -3.391164992 + -0.5 Combine like terms: -3.391164992 + -0.5 = -3.891164992 x = -3.891164992 Simplifying x = -3.891164992

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.891164992, -3.891164992}

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