6x-x^2=32-12x+x^2

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Solution for 6x-x^2=32-12x+x^2 equation:


Simplifying
6x + -1x2 = 32 + -12x + x2

Solving
6x + -1x2 = 32 + -12x + x2

Solving for variable 'x'.

Reorder the terms:
-32 + 6x + 12x + -1x2 + -1x2 = 32 + -12x + x2 + -32 + 12x + -1x2

Combine like terms: 6x + 12x = 18x
-32 + 18x + -1x2 + -1x2 = 32 + -12x + x2 + -32 + 12x + -1x2

Combine like terms: -1x2 + -1x2 = -2x2
-32 + 18x + -2x2 = 32 + -12x + x2 + -32 + 12x + -1x2

Reorder the terms:
-32 + 18x + -2x2 = 32 + -32 + -12x + 12x + x2 + -1x2

Combine like terms: 32 + -32 = 0
-32 + 18x + -2x2 = 0 + -12x + 12x + x2 + -1x2
-32 + 18x + -2x2 = -12x + 12x + x2 + -1x2

Combine like terms: -12x + 12x = 0
-32 + 18x + -2x2 = 0 + x2 + -1x2
-32 + 18x + -2x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-32 + 18x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-16 + 9x + -1x2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(-16 + 9x + -1x2)' equal to zero and attempt to solve: Simplifying -16 + 9x + -1x2 = 0 Solving -16 + 9x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 16 + -9x + x2 = 0 Move the constant term to the right: Add '-16' to each side of the equation. 16 + -9x + -16 + x2 = 0 + -16 Reorder the terms: 16 + -16 + -9x + x2 = 0 + -16 Combine like terms: 16 + -16 = 0 0 + -9x + x2 = 0 + -16 -9x + x2 = 0 + -16 Combine like terms: 0 + -16 = -16 -9x + x2 = -16 The x term is -9x. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9x + 20.25 + x2 = -16 + 20.25 Reorder the terms: 20.25 + -9x + x2 = -16 + 20.25 Combine like terms: -16 + 20.25 = 4.25 20.25 + -9x + x2 = 4.25 Factor a perfect square on the left side: (x + -4.5)(x + -4.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (x + -4.5) equal to 2.061552813 and -2.061552813.

Subproblem 1

x + -4.5 = 2.061552813 Simplifying x + -4.5 = 2.061552813 Reorder the terms: -4.5 + x = 2.061552813 Solving -4.5 + x = 2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = 2.061552813 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = 2.061552813 + 4.5 x = 2.061552813 + 4.5 Combine like terms: 2.061552813 + 4.5 = 6.561552813 x = 6.561552813 Simplifying x = 6.561552813

Subproblem 2

x + -4.5 = -2.061552813 Simplifying x + -4.5 = -2.061552813 Reorder the terms: -4.5 + x = -2.061552813 Solving -4.5 + x = -2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = -2.061552813 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = -2.061552813 + 4.5 x = -2.061552813 + 4.5 Combine like terms: -2.061552813 + 4.5 = 2.438447187 x = 2.438447187 Simplifying x = 2.438447187

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.561552813, 2.438447187}

Solution

x = {6.561552813, 2.438447187}

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