x^2+x+.25=75

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Solution for x^2+x+.25=75 equation:


Simplifying
x2 + x + 0.25 = 75

Reorder the terms:
0.25 + x + x2 = 75

Solving
0.25 + x + x2 = 75

Solving for variable 'x'.

Reorder the terms:
0.25 + -75 + x + x2 = 75 + -75

Combine like terms: 0.25 + -75 = -74.75
-74.75 + x + x2 = 75 + -75

Combine like terms: 75 + -75 = 0
-74.75 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '74.75' to each side of the equation.
-74.75 + x + 74.75 + x2 = 0 + 74.75

Reorder the terms:
-74.75 + 74.75 + x + x2 = 0 + 74.75

Combine like terms: -74.75 + 74.75 = 0.00
0.00 + x + x2 = 0 + 74.75
x + x2 = 0 + 74.75

Combine like terms: 0 + 74.75 = 74.75
x + x2 = 74.75

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.
x + 0.25 + x2 = 74.75 + 0.25

Reorder the terms:
0.25 + x + x2 = 74.75 + 0.25

Combine like terms: 74.75 + 0.25 = 75
0.25 + x + x2 = 75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 75

Calculate the square root of the right side: 8.660254038

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.660254038 and -8.660254038.

Subproblem 1

x + 0.5 = 8.660254038 Simplifying x + 0.5 = 8.660254038 Reorder the terms: 0.5 + x = 8.660254038 Solving 0.5 + x = 8.660254038 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 = 8.660254038 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.660254038 + -0.5 x = 8.660254038 + -0.5 Combine like terms: 8.660254038 + -0.5 = 8.160254038 x = 8.160254038 Simplifying x = 8.160254038

Subproblem 2

x + 0.5 = -8.660254038 Simplifying x + 0.5 = -8.660254038 Reorder the terms: 0.5 + x = -8.660254038 Solving 0.5 + x = -8.660254038 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 = -8.660254038 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.660254038 + -0.5 x = -8.660254038 + -0.5 Combine like terms: -8.660254038 + -0.5 = -9.160254038 x = -9.160254038 Simplifying x = -9.160254038

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.160254038, -9.160254038}

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