16x^2+31x=14

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


Simplifying
16x2 + 31x = 14

Reorder the terms:
31x + 16x2 = 14

Solving
31x + 16x2 = 14

Solving for variable 'x'.

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

Combine like terms: 14 + -14 = 0
-14 + 31x + 16x2 = 0

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 = 1.8134765625
0.9384765625 + 1.9375x + x2 = 1.8134765625

Factor a perfect square on the left side:
(x + 0.96875)(x + 0.96875) = 1.8134765625

Calculate the square root of the right side: 1.346653839

Break this problem into two subproblems by setting 
(x + 0.96875) equal to 1.346653839 and -1.346653839.

Subproblem 1

x + 0.96875 = 1.346653839 Simplifying x + 0.96875 = 1.346653839 Reorder the terms: 0.96875 + x = 1.346653839 Solving 0.96875 + x = 1.346653839 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 = 1.346653839 + -0.96875 Combine like terms: 0.96875 + -0.96875 = 0.00000 0.00000 + x = 1.346653839 + -0.96875 x = 1.346653839 + -0.96875 Combine like terms: 1.346653839 + -0.96875 = 0.377903839 x = 0.377903839 Simplifying x = 0.377903839

Subproblem 2

x + 0.96875 = -1.346653839 Simplifying x + 0.96875 = -1.346653839 Reorder the terms: 0.96875 + x = -1.346653839 Solving 0.96875 + x = -1.346653839 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 = -1.346653839 + -0.96875 Combine like terms: 0.96875 + -0.96875 = 0.00000 0.00000 + x = -1.346653839 + -0.96875 x = -1.346653839 + -0.96875 Combine like terms: -1.346653839 + -0.96875 = -2.315403839 x = -2.315403839 Simplifying x = -2.315403839

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.377903839, -2.315403839}

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