4r^2-7r-16=0

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Solution for 4r^2-7r-16=0 equation:


Simplifying
4r2 + -7r + -16 = 0

Reorder the terms:
-16 + -7r + 4r2 = 0

Solving
-16 + -7r + 4r2 = 0

Solving for variable 'r'.

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

Divide each side by '4'.
-4 + -1.75r + r2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -1.75r + 4 + r2 = 0 + 4

Reorder the terms:
-4 + 4 + -1.75r + r2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -1.75r + r2 = 0 + 4
-1.75r + r2 = 0 + 4

Combine like terms: 0 + 4 = 4
-1.75r + r2 = 4

The r term is -1.75r.  Take half its coefficient (-0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
-1.75r + 0.765625 + r2 = 4 + 0.765625

Reorder the terms:
0.765625 + -1.75r + r2 = 4 + 0.765625

Combine like terms: 4 + 0.765625 = 4.765625
0.765625 + -1.75r + r2 = 4.765625

Factor a perfect square on the left side:
(r + -0.875)(r + -0.875) = 4.765625

Calculate the square root of the right side: 2.18303115

Break this problem into two subproblems by setting 
(r + -0.875) equal to 2.18303115 and -2.18303115.

Subproblem 1

r + -0.875 = 2.18303115 Simplifying r + -0.875 = 2.18303115 Reorder the terms: -0.875 + r = 2.18303115 Solving -0.875 + r = 2.18303115 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + r = 2.18303115 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + r = 2.18303115 + 0.875 r = 2.18303115 + 0.875 Combine like terms: 2.18303115 + 0.875 = 3.05803115 r = 3.05803115 Simplifying r = 3.05803115

Subproblem 2

r + -0.875 = -2.18303115 Simplifying r + -0.875 = -2.18303115 Reorder the terms: -0.875 + r = -2.18303115 Solving -0.875 + r = -2.18303115 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + r = -2.18303115 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + r = -2.18303115 + 0.875 r = -2.18303115 + 0.875 Combine like terms: -2.18303115 + 0.875 = -1.30803115 r = -1.30803115 Simplifying r = -1.30803115

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.05803115, -1.30803115}

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