7x-3x^2=x^2-6

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Solution for 7x-3x^2=x^2-6 equation:


Simplifying
7x + -3x2 = x2 + -6

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

Solving
7x + -3x2 = -6 + x2

Solving for variable 'x'.

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

Combine like terms: -3x2 + -1x2 = -4x2
6 + 7x + -4x2 = -6 + x2 + 6 + -1x2

Reorder the terms:
6 + 7x + -4x2 = -6 + 6 + x2 + -1x2

Combine like terms: -6 + 6 = 0
6 + 7x + -4x2 = 0 + x2 + -1x2
6 + 7x + -4x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
6 + 7x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-1.5 + -1.75x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -1.75x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -1.75x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -1.75x + x2 = 0 + 1.5
-1.75x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-1.75x + x2 = 1.5

The x term is -1.75x.  Take half its coefficient (-0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
-1.75x + 0.765625 + x2 = 1.5 + 0.765625

Reorder the terms:
0.765625 + -1.75x + x2 = 1.5 + 0.765625

Combine like terms: 1.5 + 0.765625 = 2.265625
0.765625 + -1.75x + x2 = 2.265625

Factor a perfect square on the left side:
(x + -0.875)(x + -0.875) = 2.265625

Calculate the square root of the right side: 1.505199322

Break this problem into two subproblems by setting 
(x + -0.875) equal to 1.505199322 and -1.505199322.

Subproblem 1

x + -0.875 = 1.505199322 Simplifying x + -0.875 = 1.505199322 Reorder the terms: -0.875 + x = 1.505199322 Solving -0.875 + x = 1.505199322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = 1.505199322 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = 1.505199322 + 0.875 x = 1.505199322 + 0.875 Combine like terms: 1.505199322 + 0.875 = 2.380199322 x = 2.380199322 Simplifying x = 2.380199322

Subproblem 2

x + -0.875 = -1.505199322 Simplifying x + -0.875 = -1.505199322 Reorder the terms: -0.875 + x = -1.505199322 Solving -0.875 + x = -1.505199322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = -1.505199322 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = -1.505199322 + 0.875 x = -1.505199322 + 0.875 Combine like terms: -1.505199322 + 0.875 = -0.630199322 x = -0.630199322 Simplifying x = -0.630199322

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.380199322, -0.630199322}

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