4x^2=x^2+16x-18

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Solution for 4x^2=x^2+16x-18 equation:


Simplifying
4x2 = x2 + 16x + -18

Reorder the terms:
4x2 = -18 + 16x + x2

Solving
4x2 = -18 + 16x + x2

Solving for variable 'x'.

Reorder the terms:
18 + -16x + 4x2 + -1x2 = -18 + 16x + x2 + 18 + -16x + -1x2

Combine like terms: 4x2 + -1x2 = 3x2
18 + -16x + 3x2 = -18 + 16x + x2 + 18 + -16x + -1x2

Reorder the terms:
18 + -16x + 3x2 = -18 + 18 + 16x + -16x + x2 + -1x2

Combine like terms: -18 + 18 = 0
18 + -16x + 3x2 = 0 + 16x + -16x + x2 + -1x2
18 + -16x + 3x2 = 16x + -16x + x2 + -1x2

Combine like terms: 16x + -16x = 0
18 + -16x + 3x2 = 0 + x2 + -1x2
18 + -16x + 3x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
18 + -16x + 3x2 = 0

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

Divide each side by '3'.
6 + -5.333333333x + x2 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -5.333333333x + -6 + x2 = 0 + -6

Reorder the terms:
6 + -6 + -5.333333333x + x2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -5.333333333x + x2 = 0 + -6
-5.333333333x + x2 = 0 + -6

Combine like terms: 0 + -6 = -6
-5.333333333x + x2 = -6

The x term is -5.333333333x.  Take half its coefficient (-2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
-5.333333333x + 7.111111113 + x2 = -6 + 7.111111113

Reorder the terms:
7.111111113 + -5.333333333x + x2 = -6 + 7.111111113

Combine like terms: -6 + 7.111111113 = 1.111111113
7.111111113 + -5.333333333x + x2 = 1.111111113

Factor a perfect square on the left side:
(x + -2.666666667)(x + -2.666666667) = 1.111111113

Calculate the square root of the right side: 1.054092554

Break this problem into two subproblems by setting 
(x + -2.666666667) equal to 1.054092554 and -1.054092554.

Subproblem 1

x + -2.666666667 = 1.054092554 Simplifying x + -2.666666667 = 1.054092554 Reorder the terms: -2.666666667 + x = 1.054092554 Solving -2.666666667 + x = 1.054092554 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x = 1.054092554 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x = 1.054092554 + 2.666666667 x = 1.054092554 + 2.666666667 Combine like terms: 1.054092554 + 2.666666667 = 3.720759221 x = 3.720759221 Simplifying x = 3.720759221

Subproblem 2

x + -2.666666667 = -1.054092554 Simplifying x + -2.666666667 = -1.054092554 Reorder the terms: -2.666666667 + x = -1.054092554 Solving -2.666666667 + x = -1.054092554 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x = -1.054092554 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x = -1.054092554 + 2.666666667 x = -1.054092554 + 2.666666667 Combine like terms: -1.054092554 + 2.666666667 = 1.612574113 x = 1.612574113 Simplifying x = 1.612574113

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.720759221, 1.612574113}

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