2x^2+21x-9=0

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Solution for 2x^2+21x-9=0 equation:


Simplifying
2x2 + 21x + -9 = 0

Reorder the terms:
-9 + 21x + 2x2 = 0

Solving
-9 + 21x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-4.5 + 10.5x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 10.5x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 10.5x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 10.5x + x2 = 0 + 4.5
10.5x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
10.5x + x2 = 4.5

The x term is 10.5x.  Take half its coefficient (5.25).
Square it (27.5625) and add it to both sides.

Add '27.5625' to each side of the equation.
10.5x + 27.5625 + x2 = 4.5 + 27.5625

Reorder the terms:
27.5625 + 10.5x + x2 = 4.5 + 27.5625

Combine like terms: 4.5 + 27.5625 = 32.0625
27.5625 + 10.5x + x2 = 32.0625

Factor a perfect square on the left side:
(x + 5.25)(x + 5.25) = 32.0625

Calculate the square root of the right side: 5.662375826

Break this problem into two subproblems by setting 
(x + 5.25) equal to 5.662375826 and -5.662375826.

Subproblem 1

x + 5.25 = 5.662375826 Simplifying x + 5.25 = 5.662375826 Reorder the terms: 5.25 + x = 5.662375826 Solving 5.25 + x = 5.662375826 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + x = 5.662375826 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + x = 5.662375826 + -5.25 x = 5.662375826 + -5.25 Combine like terms: 5.662375826 + -5.25 = 0.412375826 x = 0.412375826 Simplifying x = 0.412375826

Subproblem 2

x + 5.25 = -5.662375826 Simplifying x + 5.25 = -5.662375826 Reorder the terms: 5.25 + x = -5.662375826 Solving 5.25 + x = -5.662375826 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + x = -5.662375826 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + x = -5.662375826 + -5.25 x = -5.662375826 + -5.25 Combine like terms: -5.662375826 + -5.25 = -10.912375826 x = -10.912375826 Simplifying x = -10.912375826

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.412375826, -10.912375826}

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