4r^2+19r+15=r

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Solution for 4r^2+19r+15=r equation:


Simplifying
4r2 + 19r + 15 = r

Reorder the terms:
15 + 19r + 4r2 = r

Solving
15 + 19r + 4r2 = r

Solving for variable 'r'.

Reorder the terms:
15 + 19r + -1r + 4r2 = r + -1r

Combine like terms: 19r + -1r = 18r
15 + 18r + 4r2 = r + -1r

Combine like terms: r + -1r = 0
15 + 18r + 4r2 = 0

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

Divide each side by '4'.
3.75 + 4.5r + r2 = 0

Move the constant term to the right:

Add '-3.75' to each side of the equation.
3.75 + 4.5r + -3.75 + r2 = 0 + -3.75

Reorder the terms:
3.75 + -3.75 + 4.5r + r2 = 0 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + 4.5r + r2 = 0 + -3.75
4.5r + r2 = 0 + -3.75

Combine like terms: 0 + -3.75 = -3.75
4.5r + r2 = -3.75

The r term is 4.5r.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5r + 5.0625 + r2 = -3.75 + 5.0625

Reorder the terms:
5.0625 + 4.5r + r2 = -3.75 + 5.0625

Combine like terms: -3.75 + 5.0625 = 1.3125
5.0625 + 4.5r + r2 = 1.3125

Factor a perfect square on the left side:
(r + 2.25)(r + 2.25) = 1.3125

Calculate the square root of the right side: 1.145643924

Break this problem into two subproblems by setting 
(r + 2.25) equal to 1.145643924 and -1.145643924.

Subproblem 1

r + 2.25 = 1.145643924 Simplifying r + 2.25 = 1.145643924 Reorder the terms: 2.25 + r = 1.145643924 Solving 2.25 + r = 1.145643924 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + r = 1.145643924 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + r = 1.145643924 + -2.25 r = 1.145643924 + -2.25 Combine like terms: 1.145643924 + -2.25 = -1.104356076 r = -1.104356076 Simplifying r = -1.104356076

Subproblem 2

r + 2.25 = -1.145643924 Simplifying r + 2.25 = -1.145643924 Reorder the terms: 2.25 + r = -1.145643924 Solving 2.25 + r = -1.145643924 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + r = -1.145643924 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + r = -1.145643924 + -2.25 r = -1.145643924 + -2.25 Combine like terms: -1.145643924 + -2.25 = -3.395643924 r = -3.395643924 Simplifying r = -3.395643924

Solution

The solution to the problem is based on the solutions from the subproblems. r = {-1.104356076, -3.395643924}

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