x^2+3x=420

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Solution for x^2+3x=420 equation:


Simplifying
x2 + 3x = 420

Reorder the terms:
3x + x2 = 420

Solving
3x + x2 = 420

Solving for variable 'x'.

Reorder the terms:
-420 + 3x + x2 = 420 + -420

Combine like terms: 420 + -420 = 0
-420 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '420' to each side of the equation.
-420 + 3x + 420 + x2 = 0 + 420

Reorder the terms:
-420 + 420 + 3x + x2 = 0 + 420

Combine like terms: -420 + 420 = 0
0 + 3x + x2 = 0 + 420
3x + x2 = 0 + 420

Combine like terms: 0 + 420 = 420
3x + x2 = 420

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 420 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 420 + 2.25

Combine like terms: 420 + 2.25 = 422.25
2.25 + 3x + x2 = 422.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 422.25

Calculate the square root of the right side: 20.548722588

Break this problem into two subproblems by setting 
(x + 1.5) equal to 20.548722588 and -20.548722588.

Subproblem 1

x + 1.5 = 20.548722588 Simplifying x + 1.5 = 20.548722588 Reorder the terms: 1.5 + x = 20.548722588 Solving 1.5 + x = 20.548722588 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 20.548722588 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 20.548722588 + -1.5 x = 20.548722588 + -1.5 Combine like terms: 20.548722588 + -1.5 = 19.048722588 x = 19.048722588 Simplifying x = 19.048722588

Subproblem 2

x + 1.5 = -20.548722588 Simplifying x + 1.5 = -20.548722588 Reorder the terms: 1.5 + x = -20.548722588 Solving 1.5 + x = -20.548722588 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -20.548722588 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -20.548722588 + -1.5 x = -20.548722588 + -1.5 Combine like terms: -20.548722588 + -1.5 = -22.048722588 x = -22.048722588 Simplifying x = -22.048722588

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.048722588, -22.048722588}

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