16r^2+6r-5=0

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Solution for 16r^2+6r-5=0 equation:


Simplifying
16r2 + 6r + -5 = 0

Reorder the terms:
-5 + 6r + 16r2 = 0

Solving
-5 + 6r + 16r2 = 0

Solving for variable 'r'.

Begin completing the square.  Divide all terms by
16 the coefficient of the squared term: 

Divide each side by '16'.
-0.3125 + 0.375r + r2 = 0

Move the constant term to the right:

Add '0.3125' to each side of the equation.
-0.3125 + 0.375r + 0.3125 + r2 = 0 + 0.3125

Reorder the terms:
-0.3125 + 0.3125 + 0.375r + r2 = 0 + 0.3125

Combine like terms: -0.3125 + 0.3125 = 0.0000
0.0000 + 0.375r + r2 = 0 + 0.3125
0.375r + r2 = 0 + 0.3125

Combine like terms: 0 + 0.3125 = 0.3125
0.375r + r2 = 0.3125

The r term is 0.375r.  Take half its coefficient (0.1875).
Square it (0.03515625) and add it to both sides.

Add '0.03515625' to each side of the equation.
0.375r + 0.03515625 + r2 = 0.3125 + 0.03515625

Reorder the terms:
0.03515625 + 0.375r + r2 = 0.3125 + 0.03515625

Combine like terms: 0.3125 + 0.03515625 = 0.34765625
0.03515625 + 0.375r + r2 = 0.34765625

Factor a perfect square on the left side:
(r + 0.1875)(r + 0.1875) = 0.34765625

Calculate the square root of the right side: 0.589623821

Break this problem into two subproblems by setting 
(r + 0.1875) equal to 0.589623821 and -0.589623821.

Subproblem 1

r + 0.1875 = 0.589623821 Simplifying r + 0.1875 = 0.589623821 Reorder the terms: 0.1875 + r = 0.589623821 Solving 0.1875 + r = 0.589623821 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + r = 0.589623821 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + r = 0.589623821 + -0.1875 r = 0.589623821 + -0.1875 Combine like terms: 0.589623821 + -0.1875 = 0.402123821 r = 0.402123821 Simplifying r = 0.402123821

Subproblem 2

r + 0.1875 = -0.589623821 Simplifying r + 0.1875 = -0.589623821 Reorder the terms: 0.1875 + r = -0.589623821 Solving 0.1875 + r = -0.589623821 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + r = -0.589623821 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + r = -0.589623821 + -0.1875 r = -0.589623821 + -0.1875 Combine like terms: -0.589623821 + -0.1875 = -0.777123821 r = -0.777123821 Simplifying r = -0.777123821

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.402123821, -0.777123821}

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