16x^2+9x-65=0

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Solution for 16x^2+9x-65=0 equation:


Simplifying
16x2 + 9x + -65 = 0

Reorder the terms:
-65 + 9x + 16x2 = 0

Solving
-65 + 9x + 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'.
-4.0625 + 0.5625x + x2 = 0

Move the constant term to the right:

Add '4.0625' to each side of the equation.
-4.0625 + 0.5625x + 4.0625 + x2 = 0 + 4.0625

Reorder the terms:
-4.0625 + 4.0625 + 0.5625x + x2 = 0 + 4.0625

Combine like terms: -4.0625 + 4.0625 = 0.0000
0.0000 + 0.5625x + x2 = 0 + 4.0625
0.5625x + x2 = 0 + 4.0625

Combine like terms: 0 + 4.0625 = 4.0625
0.5625x + x2 = 4.0625

The x term is 0.5625x.  Take half its coefficient (0.28125).
Square it (0.0791015625) and add it to both sides.

Add '0.0791015625' to each side of the equation.
0.5625x + 0.0791015625 + x2 = 4.0625 + 0.0791015625

Reorder the terms:
0.0791015625 + 0.5625x + x2 = 4.0625 + 0.0791015625

Combine like terms: 4.0625 + 0.0791015625 = 4.1416015625
0.0791015625 + 0.5625x + x2 = 4.1416015625

Factor a perfect square on the left side:
(x + 0.28125)(x + 0.28125) = 4.1416015625

Calculate the square root of the right side: 2.035092519

Break this problem into two subproblems by setting 
(x + 0.28125) equal to 2.035092519 and -2.035092519.

Subproblem 1

x + 0.28125 = 2.035092519 Simplifying x + 0.28125 = 2.035092519 Reorder the terms: 0.28125 + x = 2.035092519 Solving 0.28125 + x = 2.035092519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.28125' to each side of the equation. 0.28125 + -0.28125 + x = 2.035092519 + -0.28125 Combine like terms: 0.28125 + -0.28125 = 0.00000 0.00000 + x = 2.035092519 + -0.28125 x = 2.035092519 + -0.28125 Combine like terms: 2.035092519 + -0.28125 = 1.753842519 x = 1.753842519 Simplifying x = 1.753842519

Subproblem 2

x + 0.28125 = -2.035092519 Simplifying x + 0.28125 = -2.035092519 Reorder the terms: 0.28125 + x = -2.035092519 Solving 0.28125 + x = -2.035092519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.28125' to each side of the equation. 0.28125 + -0.28125 + x = -2.035092519 + -0.28125 Combine like terms: 0.28125 + -0.28125 = 0.00000 0.00000 + x = -2.035092519 + -0.28125 x = -2.035092519 + -0.28125 Combine like terms: -2.035092519 + -0.28125 = -2.316342519 x = -2.316342519 Simplifying x = -2.316342519

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.753842519, -2.316342519}

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