(9x^2-2)-(7x+5)+(3x-2)=0

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Solution for (9x^2-2)-(7x+5)+(3x-2)=0 equation:


Simplifying
(9x2 + -2) + -1(7x + 5) + (3x + -2) = 0

Reorder the terms:
(-2 + 9x2) + -1(7x + 5) + (3x + -2) = 0

Remove parenthesis around (-2 + 9x2)
-2 + 9x2 + -1(7x + 5) + (3x + -2) = 0

Reorder the terms:
-2 + 9x2 + -1(5 + 7x) + (3x + -2) = 0
-2 + 9x2 + (5 * -1 + 7x * -1) + (3x + -2) = 0
-2 + 9x2 + (-5 + -7x) + (3x + -2) = 0

Reorder the terms:
-2 + 9x2 + -5 + -7x + (-2 + 3x) = 0

Remove parenthesis around (-2 + 3x)
-2 + 9x2 + -5 + -7x + -2 + 3x = 0

Reorder the terms:
-2 + -5 + -2 + -7x + 3x + 9x2 = 0

Combine like terms: -2 + -5 = -7
-7 + -2 + -7x + 3x + 9x2 = 0

Combine like terms: -7 + -2 = -9
-9 + -7x + 3x + 9x2 = 0

Combine like terms: -7x + 3x = -4x
-9 + -4x + 9x2 = 0

Solving
-9 + -4x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
-1 + -0.4444444444x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.4444444444x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.4444444444x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.4444444444x + x2 = 0 + 1
-0.4444444444x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.4444444444x + x2 = 1

The x term is -0.4444444444x.  Take half its coefficient (-0.2222222222).
Square it (0.04938271604) and add it to both sides.

Add '0.04938271604' to each side of the equation.
-0.4444444444x + 0.04938271604 + x2 = 1 + 0.04938271604

Reorder the terms:
0.04938271604 + -0.4444444444x + x2 = 1 + 0.04938271604

Combine like terms: 1 + 0.04938271604 = 1.04938271604
0.04938271604 + -0.4444444444x + x2 = 1.04938271604

Factor a perfect square on the left side:
(x + -0.2222222222)(x + -0.2222222222) = 1.04938271604

Calculate the square root of the right side: 1.024393829

Break this problem into two subproblems by setting 
(x + -0.2222222222) equal to 1.024393829 and -1.024393829.

Subproblem 1

x + -0.2222222222 = 1.024393829 Simplifying x + -0.2222222222 = 1.024393829 Reorder the terms: -0.2222222222 + x = 1.024393829 Solving -0.2222222222 + x = 1.024393829 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + x = 1.024393829 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + x = 1.024393829 + 0.2222222222 x = 1.024393829 + 0.2222222222 Combine like terms: 1.024393829 + 0.2222222222 = 1.2466160512 x = 1.2466160512 Simplifying x = 1.2466160512

Subproblem 2

x + -0.2222222222 = -1.024393829 Simplifying x + -0.2222222222 = -1.024393829 Reorder the terms: -0.2222222222 + x = -1.024393829 Solving -0.2222222222 + x = -1.024393829 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + x = -1.024393829 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + x = -1.024393829 + 0.2222222222 x = -1.024393829 + 0.2222222222 Combine like terms: -1.024393829 + 0.2222222222 = -0.8021716068 x = -0.8021716068 Simplifying x = -0.8021716068

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.2466160512, -0.8021716068}

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