40=x^2+2.5*x+14.5

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Solution for 40=x^2+2.5*x+14.5 equation:


Simplifying
40 = x2 + 2.5x + 14.5

Reorder the terms:
40 = 14.5 + 2.5x + x2

Solving
40 = 14.5 + 2.5x + x2

Solving for variable 'x'.

Combine like terms: 40 + -14.5 = 25.5
25.5 + -2.5x + -1x2 = 14.5 + 2.5x + x2 + -14.5 + -2.5x + -1x2

Reorder the terms:
25.5 + -2.5x + -1x2 = 14.5 + -14.5 + 2.5x + -2.5x + x2 + -1x2

Combine like terms: 14.5 + -14.5 = 0.0
25.5 + -2.5x + -1x2 = 0.0 + 2.5x + -2.5x + x2 + -1x2
25.5 + -2.5x + -1x2 = 2.5x + -2.5x + x2 + -1x2

Combine like terms: 2.5x + -2.5x = 0.0
25.5 + -2.5x + -1x2 = 0.0 + x2 + -1x2
25.5 + -2.5x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
25.5 + -2.5x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-25.5 + 2.5x + x2 = 0

Move the constant term to the right:

Add '25.5' to each side of the equation.
-25.5 + 2.5x + 25.5 + x2 = 0 + 25.5

Reorder the terms:
-25.5 + 25.5 + 2.5x + x2 = 0 + 25.5

Combine like terms: -25.5 + 25.5 = 0.0
0.0 + 2.5x + x2 = 0 + 25.5
2.5x + x2 = 0 + 25.5

Combine like terms: 0 + 25.5 = 25.5
2.5x + x2 = 25.5

The x term is 2.5x.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5x + 1.5625 + x2 = 25.5 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = 25.5 + 1.5625

Combine like terms: 25.5 + 1.5625 = 27.0625
1.5625 + 2.5x + x2 = 27.0625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 27.0625

Calculate the square root of the right side: 5.202163012

Break this problem into two subproblems by setting 
(x + 1.25) equal to 5.202163012 and -5.202163012.

Subproblem 1

x + 1.25 = 5.202163012 Simplifying x + 1.25 = 5.202163012 Reorder the terms: 1.25 + x = 5.202163012 Solving 1.25 + x = 5.202163012 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = 5.202163012 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 5.202163012 + -1.25 x = 5.202163012 + -1.25 Combine like terms: 5.202163012 + -1.25 = 3.952163012 x = 3.952163012 Simplifying x = 3.952163012

Subproblem 2

x + 1.25 = -5.202163012 Simplifying x + 1.25 = -5.202163012 Reorder the terms: 1.25 + x = -5.202163012 Solving 1.25 + x = -5.202163012 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = -5.202163012 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -5.202163012 + -1.25 x = -5.202163012 + -1.25 Combine like terms: -5.202163012 + -1.25 = -6.452163012 x = -6.452163012 Simplifying x = -6.452163012

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.952163012, -6.452163012}

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