40r^2+54r-7=0

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Solution for 40r^2+54r-7=0 equation:


Simplifying
40r2 + 54r + -7 = 0

Reorder the terms:
-7 + 54r + 40r2 = 0

Solving
-7 + 54r + 40r2 = 0

Solving for variable 'r'.

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

Divide each side by '40'.
-0.175 + 1.35r + r2 = 0

Move the constant term to the right:

Add '0.175' to each side of the equation.
-0.175 + 1.35r + 0.175 + r2 = 0 + 0.175

Reorder the terms:
-0.175 + 0.175 + 1.35r + r2 = 0 + 0.175

Combine like terms: -0.175 + 0.175 = 0.000
0.000 + 1.35r + r2 = 0 + 0.175
1.35r + r2 = 0 + 0.175

Combine like terms: 0 + 0.175 = 0.175
1.35r + r2 = 0.175

The r term is 1.35r.  Take half its coefficient (0.675).
Square it (0.455625) and add it to both sides.

Add '0.455625' to each side of the equation.
1.35r + 0.455625 + r2 = 0.175 + 0.455625

Reorder the terms:
0.455625 + 1.35r + r2 = 0.175 + 0.455625

Combine like terms: 0.175 + 0.455625 = 0.630625
0.455625 + 1.35r + r2 = 0.630625

Factor a perfect square on the left side:
(r + 0.675)(r + 0.675) = 0.630625

Calculate the square root of the right side: 0.794119009

Break this problem into two subproblems by setting 
(r + 0.675) equal to 0.794119009 and -0.794119009.

Subproblem 1

r + 0.675 = 0.794119009 Simplifying r + 0.675 = 0.794119009 Reorder the terms: 0.675 + r = 0.794119009 Solving 0.675 + r = 0.794119009 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.675' to each side of the equation. 0.675 + -0.675 + r = 0.794119009 + -0.675 Combine like terms: 0.675 + -0.675 = 0.000 0.000 + r = 0.794119009 + -0.675 r = 0.794119009 + -0.675 Combine like terms: 0.794119009 + -0.675 = 0.119119009 r = 0.119119009 Simplifying r = 0.119119009

Subproblem 2

r + 0.675 = -0.794119009 Simplifying r + 0.675 = -0.794119009 Reorder the terms: 0.675 + r = -0.794119009 Solving 0.675 + r = -0.794119009 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.675' to each side of the equation. 0.675 + -0.675 + r = -0.794119009 + -0.675 Combine like terms: 0.675 + -0.675 = 0.000 0.000 + r = -0.794119009 + -0.675 r = -0.794119009 + -0.675 Combine like terms: -0.794119009 + -0.675 = -1.469119009 r = -1.469119009 Simplifying r = -1.469119009

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.119119009, -1.469119009}

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