4r^2-9r-19=0

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Solution for 4r^2-9r-19=0 equation:


Simplifying
4r2 + -9r + -19 = 0

Reorder the terms:
-19 + -9r + 4r2 = 0

Solving
-19 + -9r + 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'.
-4.75 + -2.25r + r2 = 0

Move the constant term to the right:

Add '4.75' to each side of the equation.
-4.75 + -2.25r + 4.75 + r2 = 0 + 4.75

Reorder the terms:
-4.75 + 4.75 + -2.25r + r2 = 0 + 4.75

Combine like terms: -4.75 + 4.75 = 0.00
0.00 + -2.25r + r2 = 0 + 4.75
-2.25r + r2 = 0 + 4.75

Combine like terms: 0 + 4.75 = 4.75
-2.25r + r2 = 4.75

The r term is -2.25r.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25r + 1.265625 + r2 = 4.75 + 1.265625

Reorder the terms:
1.265625 + -2.25r + r2 = 4.75 + 1.265625

Combine like terms: 4.75 + 1.265625 = 6.015625
1.265625 + -2.25r + r2 = 6.015625

Factor a perfect square on the left side:
(r + -1.125)(r + -1.125) = 6.015625

Calculate the square root of the right side: 2.452677109

Break this problem into two subproblems by setting 
(r + -1.125) equal to 2.452677109 and -2.452677109.

Subproblem 1

r + -1.125 = 2.452677109 Simplifying r + -1.125 = 2.452677109 Reorder the terms: -1.125 + r = 2.452677109 Solving -1.125 + r = 2.452677109 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + r = 2.452677109 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + r = 2.452677109 + 1.125 r = 2.452677109 + 1.125 Combine like terms: 2.452677109 + 1.125 = 3.577677109 r = 3.577677109 Simplifying r = 3.577677109

Subproblem 2

r + -1.125 = -2.452677109 Simplifying r + -1.125 = -2.452677109 Reorder the terms: -1.125 + r = -2.452677109 Solving -1.125 + r = -2.452677109 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + r = -2.452677109 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + r = -2.452677109 + 1.125 r = -2.452677109 + 1.125 Combine like terms: -2.452677109 + 1.125 = -1.327677109 r = -1.327677109 Simplifying r = -1.327677109

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.577677109, -1.327677109}

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