9x^2+4x-9=0

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Solution for 9x^2+4x-9=0 equation:


Simplifying
9x2 + 4x + -9 = 0

Reorder the terms:
-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 = 0.8021716068 x = 0.8021716068 Simplifying x = 0.8021716068

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 = -1.2466160512 x = -1.2466160512 Simplifying x = -1.2466160512

Solution

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

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