x^2+45x=1000

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Solution for x^2+45x=1000 equation:


Simplifying
x2 + 45x = 1000

Reorder the terms:
45x + x2 = 1000

Solving
45x + x2 = 1000

Solving for variable 'x'.

Reorder the terms:
-1000 + 45x + x2 = 1000 + -1000

Combine like terms: 1000 + -1000 = 0
-1000 + 45x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1000' to each side of the equation.
-1000 + 45x + 1000 + x2 = 0 + 1000

Reorder the terms:
-1000 + 1000 + 45x + x2 = 0 + 1000

Combine like terms: -1000 + 1000 = 0
0 + 45x + x2 = 0 + 1000
45x + x2 = 0 + 1000

Combine like terms: 0 + 1000 = 1000
45x + x2 = 1000

The x term is 45x.  Take half its coefficient (22.5).
Square it (506.25) and add it to both sides.

Add '506.25' to each side of the equation.
45x + 506.25 + x2 = 1000 + 506.25

Reorder the terms:
506.25 + 45x + x2 = 1000 + 506.25

Combine like terms: 1000 + 506.25 = 1506.25
506.25 + 45x + x2 = 1506.25

Factor a perfect square on the left side:
(x + 22.5)(x + 22.5) = 1506.25

Calculate the square root of the right side: 38.810436741

Break this problem into two subproblems by setting 
(x + 22.5) equal to 38.810436741 and -38.810436741.

Subproblem 1

x + 22.5 = 38.810436741 Simplifying x + 22.5 = 38.810436741 Reorder the terms: 22.5 + x = 38.810436741 Solving 22.5 + x = 38.810436741 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + x = 38.810436741 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + x = 38.810436741 + -22.5 x = 38.810436741 + -22.5 Combine like terms: 38.810436741 + -22.5 = 16.310436741 x = 16.310436741 Simplifying x = 16.310436741

Subproblem 2

x + 22.5 = -38.810436741 Simplifying x + 22.5 = -38.810436741 Reorder the terms: 22.5 + x = -38.810436741 Solving 22.5 + x = -38.810436741 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + x = -38.810436741 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + x = -38.810436741 + -22.5 x = -38.810436741 + -22.5 Combine like terms: -38.810436741 + -22.5 = -61.310436741 x = -61.310436741 Simplifying x = -61.310436741

Solution

The solution to the problem is based on the solutions from the subproblems. x = {16.310436741, -61.310436741}

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