x^2+1.5*x=1000

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Solution for x^2+1.5*x=1000 equation:


Simplifying
x2 + 1.5x = 1000

Reorder the terms:
1.5x + x2 = 1000

Solving
1.5x + x2 = 1000

Solving for variable 'x'.

Reorder the terms:
-1000 + 1.5x + x2 = 1000 + -1000

Combine like terms: 1000 + -1000 = 0
-1000 + 1.5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1000' to each side of the equation.
-1000 + 1.5x + 1000 + x2 = 0 + 1000

Reorder the terms:
-1000 + 1000 + 1.5x + x2 = 0 + 1000

Combine like terms: -1000 + 1000 = 0
0 + 1.5x + x2 = 0 + 1000
1.5x + x2 = 0 + 1000

Combine like terms: 0 + 1000 = 1000
1.5x + x2 = 1000

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 1000 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 1000 + 0.5625

Combine like terms: 1000 + 0.5625 = 1000.5625
0.5625 + 1.5x + x2 = 1000.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 1000.5625

Calculate the square root of the right side: 31.631669257

Break this problem into two subproblems by setting 
(x + 0.75) equal to 31.631669257 and -31.631669257.

Subproblem 1

x + 0.75 = 31.631669257 Simplifying x + 0.75 = 31.631669257 Reorder the terms: 0.75 + x = 31.631669257 Solving 0.75 + x = 31.631669257 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 31.631669257 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 31.631669257 + -0.75 x = 31.631669257 + -0.75 Combine like terms: 31.631669257 + -0.75 = 30.881669257 x = 30.881669257 Simplifying x = 30.881669257

Subproblem 2

x + 0.75 = -31.631669257 Simplifying x + 0.75 = -31.631669257 Reorder the terms: 0.75 + x = -31.631669257 Solving 0.75 + x = -31.631669257 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -31.631669257 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -31.631669257 + -0.75 x = -31.631669257 + -0.75 Combine like terms: -31.631669257 + -0.75 = -32.381669257 x = -32.381669257 Simplifying x = -32.381669257

Solution

The solution to the problem is based on the solutions from the subproblems. x = {30.881669257, -32.381669257}

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