x^2+4x=2x^2+7x-6

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Solution for x^2+4x=2x^2+7x-6 equation:


Simplifying
x2 + 4x = 2x2 + 7x + -6

Reorder the terms:
4x + x2 = 2x2 + 7x + -6

Reorder the terms:
4x + x2 = -6 + 7x + 2x2

Solving
4x + x2 = -6 + 7x + 2x2

Solving for variable 'x'.

Reorder the terms:
6 + 4x + -7x + x2 + -2x2 = -6 + 7x + 2x2 + 6 + -7x + -2x2

Combine like terms: 4x + -7x = -3x
6 + -3x + x2 + -2x2 = -6 + 7x + 2x2 + 6 + -7x + -2x2

Combine like terms: x2 + -2x2 = -1x2
6 + -3x + -1x2 = -6 + 7x + 2x2 + 6 + -7x + -2x2

Reorder the terms:
6 + -3x + -1x2 = -6 + 6 + 7x + -7x + 2x2 + -2x2

Combine like terms: -6 + 6 = 0
6 + -3x + -1x2 = 0 + 7x + -7x + 2x2 + -2x2
6 + -3x + -1x2 = 7x + -7x + 2x2 + -2x2

Combine like terms: 7x + -7x = 0
6 + -3x + -1x2 = 0 + 2x2 + -2x2
6 + -3x + -1x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
6 + -3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-6 + 3x + x2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 3x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + 3x + x2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 3x + x2 = 0 + 6
3x + x2 = 0 + 6

Combine like terms: 0 + 6 = 6
3x + x2 = 6

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 6 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 6 + 2.25

Combine like terms: 6 + 2.25 = 8.25
2.25 + 3x + x2 = 8.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

x + 1.5 = 2.872281323 Simplifying x + 1.5 = 2.872281323 Reorder the terms: 1.5 + x = 2.872281323 Solving 1.5 + x = 2.872281323 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 2.872281323 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 2.872281323 + -1.5 x = 2.872281323 + -1.5 Combine like terms: 2.872281323 + -1.5 = 1.372281323 x = 1.372281323 Simplifying x = 1.372281323

Subproblem 2

x + 1.5 = -2.872281323 Simplifying x + 1.5 = -2.872281323 Reorder the terms: 1.5 + x = -2.872281323 Solving 1.5 + x = -2.872281323 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -2.872281323 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -2.872281323 + -1.5 x = -2.872281323 + -1.5 Combine like terms: -2.872281323 + -1.5 = -4.372281323 x = -4.372281323 Simplifying x = -4.372281323

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.372281323, -4.372281323}

 

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