4r^2+27r+40=0

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Solution for 4r^2+27r+40=0 equation:


Simplifying
4r2 + 27r + 40 = 0

Reorder the terms:
40 + 27r + 4r2 = 0

Solving
40 + 27r + 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 + 6.75r + r2 = 0

Move the constant term to the right:

Add '-10' to each side of the equation.
10 + 6.75r + -10 + r2 = 0 + -10

Reorder the terms:
10 + -10 + 6.75r + r2 = 0 + -10

Combine like terms: 10 + -10 = 0
0 + 6.75r + r2 = 0 + -10
6.75r + r2 = 0 + -10

Combine like terms: 0 + -10 = -10
6.75r + r2 = -10

The r term is 6.75r.  Take half its coefficient (3.375).
Square it (11.390625) and add it to both sides.

Add '11.390625' to each side of the equation.
6.75r + 11.390625 + r2 = -10 + 11.390625

Reorder the terms:
11.390625 + 6.75r + r2 = -10 + 11.390625

Combine like terms: -10 + 11.390625 = 1.390625
11.390625 + 6.75r + r2 = 1.390625

Factor a perfect square on the left side:
(r + 3.375)(r + 3.375) = 1.390625

Calculate the square root of the right side: 1.179247642

Break this problem into two subproblems by setting 
(r + 3.375) equal to 1.179247642 and -1.179247642.

Subproblem 1

r + 3.375 = 1.179247642 Simplifying r + 3.375 = 1.179247642 Reorder the terms: 3.375 + r = 1.179247642 Solving 3.375 + r = 1.179247642 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.375' to each side of the equation. 3.375 + -3.375 + r = 1.179247642 + -3.375 Combine like terms: 3.375 + -3.375 = 0.000 0.000 + r = 1.179247642 + -3.375 r = 1.179247642 + -3.375 Combine like terms: 1.179247642 + -3.375 = -2.195752358 r = -2.195752358 Simplifying r = -2.195752358

Subproblem 2

r + 3.375 = -1.179247642 Simplifying r + 3.375 = -1.179247642 Reorder the terms: 3.375 + r = -1.179247642 Solving 3.375 + r = -1.179247642 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.375' to each side of the equation. 3.375 + -3.375 + r = -1.179247642 + -3.375 Combine like terms: 3.375 + -3.375 = 0.000 0.000 + r = -1.179247642 + -3.375 r = -1.179247642 + -3.375 Combine like terms: -1.179247642 + -3.375 = -4.554247642 r = -4.554247642 Simplifying r = -4.554247642

Solution

The solution to the problem is based on the solutions from the subproblems. r = {-2.195752358, -4.554247642}

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