0=16x^2-27x+7

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Solution for 0=16x^2-27x+7 equation:


Simplifying
0 = 16x2 + -27x + 7

Reorder the terms:
0 = 7 + -27x + 16x2

Solving
0 = 7 + -27x + 16x2

Solving for variable 'x'.

Combine like terms: 0 + -7 = -7
-7 + 27x + -16x2 = 7 + -27x + 16x2 + -7 + 27x + -16x2

Reorder the terms:
-7 + 27x + -16x2 = 7 + -7 + -27x + 27x + 16x2 + -16x2

Combine like terms: 7 + -7 = 0
-7 + 27x + -16x2 = 0 + -27x + 27x + 16x2 + -16x2
-7 + 27x + -16x2 = -27x + 27x + 16x2 + -16x2

Combine like terms: -27x + 27x = 0
-7 + 27x + -16x2 = 0 + 16x2 + -16x2
-7 + 27x + -16x2 = 16x2 + -16x2

Combine like terms: 16x2 + -16x2 = 0
-7 + 27x + -16x2 = 0

Begin completing the square.  Divide all terms by
-16 the coefficient of the squared term: 

Divide each side by '-16'.
0.4375 + -1.6875x + x2 = 0

Move the constant term to the right:

Add '-0.4375' to each side of the equation.
0.4375 + -1.6875x + -0.4375 + x2 = 0 + -0.4375

Reorder the terms:
0.4375 + -0.4375 + -1.6875x + x2 = 0 + -0.4375

Combine like terms: 0.4375 + -0.4375 = 0.0000
0.0000 + -1.6875x + x2 = 0 + -0.4375
-1.6875x + x2 = 0 + -0.4375

Combine like terms: 0 + -0.4375 = -0.4375
-1.6875x + x2 = -0.4375

The x term is -1.6875x.  Take half its coefficient (-0.84375).
Square it (0.7119140625) and add it to both sides.

Add '0.7119140625' to each side of the equation.
-1.6875x + 0.7119140625 + x2 = -0.4375 + 0.7119140625

Reorder the terms:
0.7119140625 + -1.6875x + x2 = -0.4375 + 0.7119140625

Combine like terms: -0.4375 + 0.7119140625 = 0.2744140625
0.7119140625 + -1.6875x + x2 = 0.2744140625

Factor a perfect square on the left side:
(x + -0.84375)(x + -0.84375) = 0.2744140625

Calculate the square root of the right side: 0.523845457

Break this problem into two subproblems by setting 
(x + -0.84375) equal to 0.523845457 and -0.523845457.

Subproblem 1

x + -0.84375 = 0.523845457 Simplifying x + -0.84375 = 0.523845457 Reorder the terms: -0.84375 + x = 0.523845457 Solving -0.84375 + x = 0.523845457 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + x = 0.523845457 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + x = 0.523845457 + 0.84375 x = 0.523845457 + 0.84375 Combine like terms: 0.523845457 + 0.84375 = 1.367595457 x = 1.367595457 Simplifying x = 1.367595457

Subproblem 2

x + -0.84375 = -0.523845457 Simplifying x + -0.84375 = -0.523845457 Reorder the terms: -0.84375 + x = -0.523845457 Solving -0.84375 + x = -0.523845457 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + x = -0.523845457 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + x = -0.523845457 + 0.84375 x = -0.523845457 + 0.84375 Combine like terms: -0.523845457 + 0.84375 = 0.319904543 x = 0.319904543 Simplifying x = 0.319904543

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.367595457, 0.319904543}

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