9x^2-4x+2=53

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Solution for 9x^2-4x+2=53 equation:


Simplifying
9x2 + -4x + 2 = 53

Reorder the terms:
2 + -4x + 9x2 = 53

Solving
2 + -4x + 9x2 = 53

Solving for variable 'x'.

Reorder the terms:
2 + -53 + -4x + 9x2 = 53 + -53

Combine like terms: 2 + -53 = -51
-51 + -4x + 9x2 = 53 + -53

Combine like terms: 53 + -53 = 0
-51 + -4x + 9x2 = 0

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

Divide each side by '9'.
-5.666666667 + -0.4444444444x + x2 = 0

Move the constant term to the right:

Add '5.666666667' to each side of the equation.
-5.666666667 + -0.4444444444x + 5.666666667 + x2 = 0 + 5.666666667

Reorder the terms:
-5.666666667 + 5.666666667 + -0.4444444444x + x2 = 0 + 5.666666667

Combine like terms: -5.666666667 + 5.666666667 = 0.000000000
0.000000000 + -0.4444444444x + x2 = 0 + 5.666666667
-0.4444444444x + x2 = 0 + 5.666666667

Combine like terms: 0 + 5.666666667 = 5.666666667
-0.4444444444x + x2 = 5.666666667

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 = 5.666666667 + 0.04938271604

Reorder the terms:
0.04938271604 + -0.4444444444x + x2 = 5.666666667 + 0.04938271604

Combine like terms: 5.666666667 + 0.04938271604 = 5.71604938304
0.04938271604 + -0.4444444444x + x2 = 5.71604938304

Factor a perfect square on the left side:
(x + -0.2222222222)(x + -0.2222222222) = 5.71604938304

Calculate the square root of the right side: 2.390826088

Break this problem into two subproblems by setting 
(x + -0.2222222222) equal to 2.390826088 and -2.390826088.

Subproblem 1

x + -0.2222222222 = 2.390826088 Simplifying x + -0.2222222222 = 2.390826088 Reorder the terms: -0.2222222222 + x = 2.390826088 Solving -0.2222222222 + x = 2.390826088 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 = 2.390826088 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + x = 2.390826088 + 0.2222222222 x = 2.390826088 + 0.2222222222 Combine like terms: 2.390826088 + 0.2222222222 = 2.6130483102 x = 2.6130483102 Simplifying x = 2.6130483102

Subproblem 2

x + -0.2222222222 = -2.390826088 Simplifying x + -0.2222222222 = -2.390826088 Reorder the terms: -0.2222222222 + x = -2.390826088 Solving -0.2222222222 + x = -2.390826088 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 = -2.390826088 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + x = -2.390826088 + 0.2222222222 x = -2.390826088 + 0.2222222222 Combine like terms: -2.390826088 + 0.2222222222 = -2.1686038658 x = -2.1686038658 Simplifying x = -2.1686038658

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.6130483102, -2.1686038658}

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