16x^2-31x+14=0

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Solution for 16x^2-31x+14=0 equation:


Simplifying
16x2 + -31x + 14 = 0

Reorder the terms:
14 + -31x + 16x2 = 0

Solving
14 + -31x + 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'.
0.875 + -1.9375x + x2 = 0

Move the constant term to the right:

Add '-0.875' to each side of the equation.
0.875 + -1.9375x + -0.875 + x2 = 0 + -0.875

Reorder the terms:
0.875 + -0.875 + -1.9375x + x2 = 0 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + -1.9375x + x2 = 0 + -0.875
-1.9375x + x2 = 0 + -0.875

Combine like terms: 0 + -0.875 = -0.875
-1.9375x + x2 = -0.875

The x term is -1.9375x.  Take half its coefficient (-0.96875).
Square it (0.9384765625) and add it to both sides.

Add '0.9384765625' to each side of the equation.
-1.9375x + 0.9384765625 + x2 = -0.875 + 0.9384765625

Reorder the terms:
0.9384765625 + -1.9375x + x2 = -0.875 + 0.9384765625

Combine like terms: -0.875 + 0.9384765625 = 0.0634765625
0.9384765625 + -1.9375x + x2 = 0.0634765625

Factor a perfect square on the left side:
(x + -0.96875)(x + -0.96875) = 0.0634765625

Calculate the square root of the right side: 0.251945555

Break this problem into two subproblems by setting 
(x + -0.96875) equal to 0.251945555 and -0.251945555.

Subproblem 1

x + -0.96875 = 0.251945555 Simplifying x + -0.96875 = 0.251945555 Reorder the terms: -0.96875 + x = 0.251945555 Solving -0.96875 + x = 0.251945555 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + x = 0.251945555 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + x = 0.251945555 + 0.96875 x = 0.251945555 + 0.96875 Combine like terms: 0.251945555 + 0.96875 = 1.220695555 x = 1.220695555 Simplifying x = 1.220695555

Subproblem 2

x + -0.96875 = -0.251945555 Simplifying x + -0.96875 = -0.251945555 Reorder the terms: -0.96875 + x = -0.251945555 Solving -0.96875 + x = -0.251945555 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + x = -0.251945555 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + x = -0.251945555 + 0.96875 x = -0.251945555 + 0.96875 Combine like terms: -0.251945555 + 0.96875 = 0.716804445 x = 0.716804445 Simplifying x = 0.716804445

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.220695555, 0.716804445}

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