16x^2+78x-27=0

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Solution for 16x^2+78x-27=0 equation:


Simplifying
16x2 + 78x + -27 = 0

Reorder the terms:
-27 + 78x + 16x2 = 0

Solving
-27 + 78x + 16x2 = 0

Solving for variable 'x'.

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

Divide each side by '16'.
-1.6875 + 4.875x + x2 = 0

Move the constant term to the right:

Add '1.6875' to each side of the equation.
-1.6875 + 4.875x + 1.6875 + x2 = 0 + 1.6875

Reorder the terms:
-1.6875 + 1.6875 + 4.875x + x2 = 0 + 1.6875

Combine like terms: -1.6875 + 1.6875 = 0.0000
0.0000 + 4.875x + x2 = 0 + 1.6875
4.875x + x2 = 0 + 1.6875

Combine like terms: 0 + 1.6875 = 1.6875
4.875x + x2 = 1.6875

The x term is 4.875x.  Take half its coefficient (2.4375).
Square it (5.94140625) and add it to both sides.

Add '5.94140625' to each side of the equation.
4.875x + 5.94140625 + x2 = 1.6875 + 5.94140625

Reorder the terms:
5.94140625 + 4.875x + x2 = 1.6875 + 5.94140625

Combine like terms: 1.6875 + 5.94140625 = 7.62890625
5.94140625 + 4.875x + x2 = 7.62890625

Factor a perfect square on the left side:
(x + 2.4375)(x + 2.4375) = 7.62890625

Calculate the square root of the right side: 2.762047474

Break this problem into two subproblems by setting 
(x + 2.4375) equal to 2.762047474 and -2.762047474.

Subproblem 1

x + 2.4375 = 2.762047474 Simplifying x + 2.4375 = 2.762047474 Reorder the terms: 2.4375 + x = 2.762047474 Solving 2.4375 + x = 2.762047474 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.4375' to each side of the equation. 2.4375 + -2.4375 + x = 2.762047474 + -2.4375 Combine like terms: 2.4375 + -2.4375 = 0.0000 0.0000 + x = 2.762047474 + -2.4375 x = 2.762047474 + -2.4375 Combine like terms: 2.762047474 + -2.4375 = 0.324547474 x = 0.324547474 Simplifying x = 0.324547474

Subproblem 2

x + 2.4375 = -2.762047474 Simplifying x + 2.4375 = -2.762047474 Reorder the terms: 2.4375 + x = -2.762047474 Solving 2.4375 + x = -2.762047474 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.4375' to each side of the equation. 2.4375 + -2.4375 + x = -2.762047474 + -2.4375 Combine like terms: 2.4375 + -2.4375 = 0.0000 0.0000 + x = -2.762047474 + -2.4375 x = -2.762047474 + -2.4375 Combine like terms: -2.762047474 + -2.4375 = -5.199547474 x = -5.199547474 Simplifying x = -5.199547474

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.324547474, -5.199547474}

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