x^2+14x=-x^2+3x+5

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Solution for x^2+14x=-x^2+3x+5 equation:


Simplifying
x2 + 14x = -1x2 + 3x + 5

Reorder the terms:
14x + x2 = -1x2 + 3x + 5

Reorder the terms:
14x + x2 = 5 + 3x + -1x2

Solving
14x + x2 = 5 + 3x + -1x2

Solving for variable 'x'.

Reorder the terms:
-5 + 14x + -3x + x2 + x2 = 5 + 3x + -1x2 + -5 + -3x + x2

Combine like terms: 14x + -3x = 11x
-5 + 11x + x2 + x2 = 5 + 3x + -1x2 + -5 + -3x + x2

Combine like terms: x2 + x2 = 2x2
-5 + 11x + 2x2 = 5 + 3x + -1x2 + -5 + -3x + x2

Reorder the terms:
-5 + 11x + 2x2 = 5 + -5 + 3x + -3x + -1x2 + x2

Combine like terms: 5 + -5 = 0
-5 + 11x + 2x2 = 0 + 3x + -3x + -1x2 + x2
-5 + 11x + 2x2 = 3x + -3x + -1x2 + x2

Combine like terms: 3x + -3x = 0
-5 + 11x + 2x2 = 0 + -1x2 + x2
-5 + 11x + 2x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
-5 + 11x + 2x2 = 0

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

Divide each side by '2'.
-2.5 + 5.5x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 5.5x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 5.5x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 5.5x + x2 = 0 + 2.5
5.5x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
5.5x + x2 = 2.5

The x term is 5.5x.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5x + 7.5625 + x2 = 2.5 + 7.5625

Reorder the terms:
7.5625 + 5.5x + x2 = 2.5 + 7.5625

Combine like terms: 2.5 + 7.5625 = 10.0625
7.5625 + 5.5x + x2 = 10.0625

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 10.0625

Calculate the square root of the right side: 3.172144385

Break this problem into two subproblems by setting 
(x + 2.75) equal to 3.172144385 and -3.172144385.

Subproblem 1

x + 2.75 = 3.172144385 Simplifying x + 2.75 = 3.172144385 Reorder the terms: 2.75 + x = 3.172144385 Solving 2.75 + x = 3.172144385 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = 3.172144385 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = 3.172144385 + -2.75 x = 3.172144385 + -2.75 Combine like terms: 3.172144385 + -2.75 = 0.422144385 x = 0.422144385 Simplifying x = 0.422144385

Subproblem 2

x + 2.75 = -3.172144385 Simplifying x + 2.75 = -3.172144385 Reorder the terms: 2.75 + x = -3.172144385 Solving 2.75 + x = -3.172144385 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = -3.172144385 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = -3.172144385 + -2.75 x = -3.172144385 + -2.75 Combine like terms: -3.172144385 + -2.75 = -5.922144385 x = -5.922144385 Simplifying x = -5.922144385

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.422144385, -5.922144385}

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