9x^2+16x-9=0

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Solution for 9x^2+16x-9=0 equation:


Simplifying
9x2 + 16x + -9 = 0

Reorder the terms:
-9 + 16x + 9x2 = 0

Solving
-9 + 16x + 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 + 1.777777778x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.777777778x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.777777778x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.777777778x + x2 = 0 + 1
1.777777778x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.777777778x + x2 = 1

The x term is 1.777777778x.  Take half its coefficient (0.888888889).
Square it (0.7901234570) and add it to both sides.

Add '0.7901234570' to each side of the equation.
1.777777778x + 0.7901234570 + x2 = 1 + 0.7901234570

Reorder the terms:
0.7901234570 + 1.777777778x + x2 = 1 + 0.7901234570

Combine like terms: 1 + 0.7901234570 = 1.790123457
0.7901234570 + 1.777777778x + x2 = 1.790123457

Factor a perfect square on the left side:
(x + 0.888888889)(x + 0.888888889) = 1.790123457

Calculate the square root of the right side: 1.337954953

Break this problem into two subproblems by setting 
(x + 0.888888889) equal to 1.337954953 and -1.337954953.

Subproblem 1

x + 0.888888889 = 1.337954953 Simplifying x + 0.888888889 = 1.337954953 Reorder the terms: 0.888888889 + x = 1.337954953 Solving 0.888888889 + x = 1.337954953 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.888888889' to each side of the equation. 0.888888889 + -0.888888889 + x = 1.337954953 + -0.888888889 Combine like terms: 0.888888889 + -0.888888889 = 0.000000000 0.000000000 + x = 1.337954953 + -0.888888889 x = 1.337954953 + -0.888888889 Combine like terms: 1.337954953 + -0.888888889 = 0.449066064 x = 0.449066064 Simplifying x = 0.449066064

Subproblem 2

x + 0.888888889 = -1.337954953 Simplifying x + 0.888888889 = -1.337954953 Reorder the terms: 0.888888889 + x = -1.337954953 Solving 0.888888889 + x = -1.337954953 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.888888889' to each side of the equation. 0.888888889 + -0.888888889 + x = -1.337954953 + -0.888888889 Combine like terms: 0.888888889 + -0.888888889 = 0.000000000 0.000000000 + x = -1.337954953 + -0.888888889 x = -1.337954953 + -0.888888889 Combine like terms: -1.337954953 + -0.888888889 = -2.226843842 x = -2.226843842 Simplifying x = -2.226843842

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.449066064, -2.226843842}

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