0=15x^2+22.5x-405

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Solution for 0=15x^2+22.5x-405 equation:


Simplifying
0 = 15x2 + 22.5x + -405

Reorder the terms:
0 = -405 + 22.5x + 15x2

Solving
0 = -405 + 22.5x + 15x2

Solving for variable 'x'.

Combine like terms: 0 + 405 = 405
405 + -22.5x + -15x2 = -405 + 22.5x + 15x2 + 405 + -22.5x + -15x2

Reorder the terms:
405 + -22.5x + -15x2 = -405 + 405 + 22.5x + -22.5x + 15x2 + -15x2

Combine like terms: -405 + 405 = 0
405 + -22.5x + -15x2 = 0 + 22.5x + -22.5x + 15x2 + -15x2
405 + -22.5x + -15x2 = 22.5x + -22.5x + 15x2 + -15x2

Combine like terms: 22.5x + -22.5x = 0.0
405 + -22.5x + -15x2 = 0.0 + 15x2 + -15x2
405 + -22.5x + -15x2 = 15x2 + -15x2

Combine like terms: 15x2 + -15x2 = 0
405 + -22.5x + -15x2 = 0

Begin completing the square.  Divide all terms by
-15 the coefficient of the squared term: 

Divide each side by '-15'.
-27 + 1.5x + x2 = 0

Move the constant term to the right:

Add '27' to each side of the equation.
-27 + 1.5x + 27 + x2 = 0 + 27

Reorder the terms:
-27 + 27 + 1.5x + x2 = 0 + 27

Combine like terms: -27 + 27 = 0
0 + 1.5x + x2 = 0 + 27
1.5x + x2 = 0 + 27

Combine like terms: 0 + 27 = 27
1.5x + x2 = 27

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 = 27 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 27 + 0.5625

Combine like terms: 27 + 0.5625 = 27.5625
0.5625 + 1.5x + x2 = 27.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 27.5625

Calculate the square root of the right side: 5.25

Break this problem into two subproblems by setting 
(x + 0.75) equal to 5.25 and -5.25.

Subproblem 1

x + 0.75 = 5.25 Simplifying x + 0.75 = 5.25 Reorder the terms: 0.75 + x = 5.25 Solving 0.75 + x = 5.25 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 = 5.25 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 5.25 + -0.75 x = 5.25 + -0.75 Combine like terms: 5.25 + -0.75 = 4.5 x = 4.5 Simplifying x = 4.5

Subproblem 2

x + 0.75 = -5.25 Simplifying x + 0.75 = -5.25 Reorder the terms: 0.75 + x = -5.25 Solving 0.75 + x = -5.25 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 = -5.25 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -5.25 + -0.75 x = -5.25 + -0.75 Combine like terms: -5.25 + -0.75 = -6 x = -6 Simplifying x = -6

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.5, -6}

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