9x^2-16x+3=0

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Solution for 9x^2-16x+3=0 equation:


Simplifying
9x2 + -16x + 3 = 0

Reorder the terms:
3 + -16x + 9x2 = 0

Solving
3 + -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'.
0.3333333333 + -1.777777778x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -1.777777778x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -1.777777778x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -1.777777778x + x2 = 0 + -0.3333333333
-1.777777778x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-1.777777778x + x2 = -0.3333333333

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 = -0.3333333333 + 0.7901234570

Reorder the terms:
0.7901234570 + -1.777777778x + x2 = -0.3333333333 + 0.7901234570

Combine like terms: -0.3333333333 + 0.7901234570 = 0.4567901237
0.7901234570 + -1.777777778x + x2 = 0.4567901237

Factor a perfect square on the left side:
(x + -0.888888889)(x + -0.888888889) = 0.4567901237

Calculate the square root of the right side: 0.675862504

Break this problem into two subproblems by setting 
(x + -0.888888889) equal to 0.675862504 and -0.675862504.

Subproblem 1

x + -0.888888889 = 0.675862504 Simplifying x + -0.888888889 = 0.675862504 Reorder the terms: -0.888888889 + x = 0.675862504 Solving -0.888888889 + x = 0.675862504 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 = 0.675862504 + 0.888888889 Combine like terms: -0.888888889 + 0.888888889 = 0.000000000 0.000000000 + x = 0.675862504 + 0.888888889 x = 0.675862504 + 0.888888889 Combine like terms: 0.675862504 + 0.888888889 = 1.564751393 x = 1.564751393 Simplifying x = 1.564751393

Subproblem 2

x + -0.888888889 = -0.675862504 Simplifying x + -0.888888889 = -0.675862504 Reorder the terms: -0.888888889 + x = -0.675862504 Solving -0.888888889 + x = -0.675862504 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 = -0.675862504 + 0.888888889 Combine like terms: -0.888888889 + 0.888888889 = 0.000000000 0.000000000 + x = -0.675862504 + 0.888888889 x = -0.675862504 + 0.888888889 Combine like terms: -0.675862504 + 0.888888889 = 0.213026385 x = 0.213026385 Simplifying x = 0.213026385

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.564751393, 0.213026385}

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