16x^2+31x-15=0

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Solution for 16x^2+31x-15=0 equation:


Simplifying
16x2 + 31x + -15 = 0

Reorder the terms:
-15 + 31x + 16x2 = 0

Solving
-15 + 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'.
-0.9375 + 1.9375x + x2 = 0

Move the constant term to the right:

Add '0.9375' to each side of the equation.
-0.9375 + 1.9375x + 0.9375 + x2 = 0 + 0.9375

Reorder the terms:
-0.9375 + 0.9375 + 1.9375x + x2 = 0 + 0.9375

Combine like terms: -0.9375 + 0.9375 = 0.0000
0.0000 + 1.9375x + x2 = 0 + 0.9375
1.9375x + x2 = 0 + 0.9375

Combine like terms: 0 + 0.9375 = 0.9375
1.9375x + x2 = 0.9375

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 = 0.9375 + 0.9384765625

Reorder the terms:
0.9384765625 + 1.9375x + x2 = 0.9375 + 0.9384765625

Combine like terms: 0.9375 + 0.9384765625 = 1.8759765625
0.9384765625 + 1.9375x + x2 = 1.8759765625

Factor a perfect square on the left side:
(x + 0.96875)(x + 0.96875) = 1.8759765625

Calculate the square root of the right side: 1.369662938

Break this problem into two subproblems by setting 
(x + 0.96875) equal to 1.369662938 and -1.369662938.

Subproblem 1

x + 0.96875 = 1.369662938 Simplifying x + 0.96875 = 1.369662938 Reorder the terms: 0.96875 + x = 1.369662938 Solving 0.96875 + x = 1.369662938 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 = 1.369662938 + -0.96875 Combine like terms: 0.96875 + -0.96875 = 0.00000 0.00000 + x = 1.369662938 + -0.96875 x = 1.369662938 + -0.96875 Combine like terms: 1.369662938 + -0.96875 = 0.400912938 x = 0.400912938 Simplifying x = 0.400912938

Subproblem 2

x + 0.96875 = -1.369662938 Simplifying x + 0.96875 = -1.369662938 Reorder the terms: 0.96875 + x = -1.369662938 Solving 0.96875 + x = -1.369662938 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 = -1.369662938 + -0.96875 Combine like terms: 0.96875 + -0.96875 = 0.00000 0.00000 + x = -1.369662938 + -0.96875 x = -1.369662938 + -0.96875 Combine like terms: -1.369662938 + -0.96875 = -2.338412938 x = -2.338412938 Simplifying x = -2.338412938

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.400912938, -2.338412938}

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