4r^2-3r-2=0

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Solution for 4r^2-3r-2=0 equation:


Simplifying
4r2 + -3r + -2 = 0

Reorder the terms:
-2 + -3r + 4r2 = 0

Solving
-2 + -3r + 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'.
-0.5 + -0.75r + r2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.75r + 0.5 + r2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.75r + r2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.75r + r2 = 0 + 0.5
-0.75r + r2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.75r + r2 = 0.5

The r term is -0.75r.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75r + 0.140625 + r2 = 0.5 + 0.140625

Reorder the terms:
0.140625 + -0.75r + r2 = 0.5 + 0.140625

Combine like terms: 0.5 + 0.140625 = 0.640625
0.140625 + -0.75r + r2 = 0.640625

Factor a perfect square on the left side:
(r + -0.375)(r + -0.375) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(r + -0.375) equal to 0.80039053 and -0.80039053.

Subproblem 1

r + -0.375 = 0.80039053 Simplifying r + -0.375 = 0.80039053 Reorder the terms: -0.375 + r = 0.80039053 Solving -0.375 + r = 0.80039053 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + r = 0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + r = 0.80039053 + 0.375 r = 0.80039053 + 0.375 Combine like terms: 0.80039053 + 0.375 = 1.17539053 r = 1.17539053 Simplifying r = 1.17539053

Subproblem 2

r + -0.375 = -0.80039053 Simplifying r + -0.375 = -0.80039053 Reorder the terms: -0.375 + r = -0.80039053 Solving -0.375 + r = -0.80039053 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + r = -0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + r = -0.80039053 + 0.375 r = -0.80039053 + 0.375 Combine like terms: -0.80039053 + 0.375 = -0.42539053 r = -0.42539053 Simplifying r = -0.42539053

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.17539053, -0.42539053}

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