4r^2+6r-45=0

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Solution for 4r^2+6r-45=0 equation:


Simplifying
4r2 + 6r + -45 = 0

Reorder the terms:
-45 + 6r + 4r2 = 0

Solving
-45 + 6r + 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'.
-11.25 + 1.5r + r2 = 0

Move the constant term to the right:

Add '11.25' to each side of the equation.
-11.25 + 1.5r + 11.25 + r2 = 0 + 11.25

Reorder the terms:
-11.25 + 11.25 + 1.5r + r2 = 0 + 11.25

Combine like terms: -11.25 + 11.25 = 0.00
0.00 + 1.5r + r2 = 0 + 11.25
1.5r + r2 = 0 + 11.25

Combine like terms: 0 + 11.25 = 11.25
1.5r + r2 = 11.25

The r term is 1.5r.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5r + 0.5625 + r2 = 11.25 + 0.5625

Reorder the terms:
0.5625 + 1.5r + r2 = 11.25 + 0.5625

Combine like terms: 11.25 + 0.5625 = 11.8125
0.5625 + 1.5r + r2 = 11.8125

Factor a perfect square on the left side:
(r + 0.75)(r + 0.75) = 11.8125

Calculate the square root of the right side: 3.436931771

Break this problem into two subproblems by setting 
(r + 0.75) equal to 3.436931771 and -3.436931771.

Subproblem 1

r + 0.75 = 3.436931771 Simplifying r + 0.75 = 3.436931771 Reorder the terms: 0.75 + r = 3.436931771 Solving 0.75 + r = 3.436931771 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + r = 3.436931771 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = 3.436931771 + -0.75 r = 3.436931771 + -0.75 Combine like terms: 3.436931771 + -0.75 = 2.686931771 r = 2.686931771 Simplifying r = 2.686931771

Subproblem 2

r + 0.75 = -3.436931771 Simplifying r + 0.75 = -3.436931771 Reorder the terms: 0.75 + r = -3.436931771 Solving 0.75 + r = -3.436931771 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + r = -3.436931771 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = -3.436931771 + -0.75 r = -3.436931771 + -0.75 Combine like terms: -3.436931771 + -0.75 = -4.186931771 r = -4.186931771 Simplifying r = -4.186931771

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.686931771, -4.186931771}

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