4r^2-23r-40=0

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Solution for 4r^2-23r-40=0 equation:


Simplifying
4r2 + -23r + -40 = 0

Reorder the terms:
-40 + -23r + 4r2 = 0

Solving
-40 + -23r + 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'.
-10 + -5.75r + r2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -5.75r + 10 + r2 = 0 + 10

Reorder the terms:
-10 + 10 + -5.75r + r2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -5.75r + r2 = 0 + 10
-5.75r + r2 = 0 + 10

Combine like terms: 0 + 10 = 10
-5.75r + r2 = 10

The r term is -5.75r.  Take half its coefficient (-2.875).
Square it (8.265625) and add it to both sides.

Add '8.265625' to each side of the equation.
-5.75r + 8.265625 + r2 = 10 + 8.265625

Reorder the terms:
8.265625 + -5.75r + r2 = 10 + 8.265625

Combine like terms: 10 + 8.265625 = 18.265625
8.265625 + -5.75r + r2 = 18.265625

Factor a perfect square on the left side:
(r + -2.875)(r + -2.875) = 18.265625

Calculate the square root of the right side: 4.273830249

Break this problem into two subproblems by setting 
(r + -2.875) equal to 4.273830249 and -4.273830249.

Subproblem 1

r + -2.875 = 4.273830249 Simplifying r + -2.875 = 4.273830249 Reorder the terms: -2.875 + r = 4.273830249 Solving -2.875 + r = 4.273830249 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + r = 4.273830249 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + r = 4.273830249 + 2.875 r = 4.273830249 + 2.875 Combine like terms: 4.273830249 + 2.875 = 7.148830249 r = 7.148830249 Simplifying r = 7.148830249

Subproblem 2

r + -2.875 = -4.273830249 Simplifying r + -2.875 = -4.273830249 Reorder the terms: -2.875 + r = -4.273830249 Solving -2.875 + r = -4.273830249 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + r = -4.273830249 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + r = -4.273830249 + 2.875 r = -4.273830249 + 2.875 Combine like terms: -4.273830249 + 2.875 = -1.398830249 r = -1.398830249 Simplifying r = -1.398830249

Solution

The solution to the problem is based on the solutions from the subproblems. r = {7.148830249, -1.398830249}

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