5r^2+7r=5

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Solution for 5r^2+7r=5 equation:


Simplifying
5r2 + 7r = 5

Reorder the terms:
7r + 5r2 = 5

Solving
7r + 5r2 = 5

Solving for variable 'r'.

Reorder the terms:
-5 + 7r + 5r2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + 7r + 5r2 = 0

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

Divide each side by '5'.
-1 + 1.4r + r2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.4r + 1 + r2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.4r + r2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.4r + r2 = 0 + 1
1.4r + r2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.4r + r2 = 1

The r term is 1.4r.  Take half its coefficient (0.7).
Square it (0.49) and add it to both sides.

Add '0.49' to each side of the equation.
1.4r + 0.49 + r2 = 1 + 0.49

Reorder the terms:
0.49 + 1.4r + r2 = 1 + 0.49

Combine like terms: 1 + 0.49 = 1.49
0.49 + 1.4r + r2 = 1.49

Factor a perfect square on the left side:
(r + 0.7)(r + 0.7) = 1.49

Calculate the square root of the right side: 1.220655562

Break this problem into two subproblems by setting 
(r + 0.7) equal to 1.220655562 and -1.220655562.

Subproblem 1

r + 0.7 = 1.220655562 Simplifying r + 0.7 = 1.220655562 Reorder the terms: 0.7 + r = 1.220655562 Solving 0.7 + r = 1.220655562 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + r = 1.220655562 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + r = 1.220655562 + -0.7 r = 1.220655562 + -0.7 Combine like terms: 1.220655562 + -0.7 = 0.520655562 r = 0.520655562 Simplifying r = 0.520655562

Subproblem 2

r + 0.7 = -1.220655562 Simplifying r + 0.7 = -1.220655562 Reorder the terms: 0.7 + r = -1.220655562 Solving 0.7 + r = -1.220655562 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + r = -1.220655562 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + r = -1.220655562 + -0.7 r = -1.220655562 + -0.7 Combine like terms: -1.220655562 + -0.7 = -1.920655562 r = -1.920655562 Simplifying r = -1.920655562

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.520655562, -1.920655562}

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