16x^2+22x-27=0

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Solution for 16x^2+22x-27=0 equation:


Simplifying
16x2 + 22x + -27 = 0

Reorder the terms:
-27 + 22x + 16x2 = 0

Solving
-27 + 22x + 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 + 1.375x + x2 = 0

Move the constant term to the right:

Add '1.6875' to each side of the equation.
-1.6875 + 1.375x + 1.6875 + x2 = 0 + 1.6875

Reorder the terms:
-1.6875 + 1.6875 + 1.375x + x2 = 0 + 1.6875

Combine like terms: -1.6875 + 1.6875 = 0.0000
0.0000 + 1.375x + x2 = 0 + 1.6875
1.375x + x2 = 0 + 1.6875

Combine like terms: 0 + 1.6875 = 1.6875
1.375x + x2 = 1.6875

The x term is 1.375x.  Take half its coefficient (0.6875).
Square it (0.47265625) and add it to both sides.

Add '0.47265625' to each side of the equation.
1.375x + 0.47265625 + x2 = 1.6875 + 0.47265625

Reorder the terms:
0.47265625 + 1.375x + x2 = 1.6875 + 0.47265625

Combine like terms: 1.6875 + 0.47265625 = 2.16015625
0.47265625 + 1.375x + x2 = 2.16015625

Factor a perfect square on the left side:
(x + 0.6875)(x + 0.6875) = 2.16015625

Calculate the square root of the right side: 1.469747002

Break this problem into two subproblems by setting 
(x + 0.6875) equal to 1.469747002 and -1.469747002.

Subproblem 1

x + 0.6875 = 1.469747002 Simplifying x + 0.6875 = 1.469747002 Reorder the terms: 0.6875 + x = 1.469747002 Solving 0.6875 + x = 1.469747002 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + x = 1.469747002 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + x = 1.469747002 + -0.6875 x = 1.469747002 + -0.6875 Combine like terms: 1.469747002 + -0.6875 = 0.782247002 x = 0.782247002 Simplifying x = 0.782247002

Subproblem 2

x + 0.6875 = -1.469747002 Simplifying x + 0.6875 = -1.469747002 Reorder the terms: 0.6875 + x = -1.469747002 Solving 0.6875 + x = -1.469747002 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + x = -1.469747002 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + x = -1.469747002 + -0.6875 x = -1.469747002 + -0.6875 Combine like terms: -1.469747002 + -0.6875 = -2.157247002 x = -2.157247002 Simplifying x = -2.157247002

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.782247002, -2.157247002}

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