12r^2+5r-12=0

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Solution for 12r^2+5r-12=0 equation:


Simplifying
12r2 + 5r + -12 = 0

Reorder the terms:
-12 + 5r + 12r2 = 0

Solving
-12 + 5r + 12r2 = 0

Solving for variable 'r'.

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

Divide each side by '12'.
-1 + 0.4166666667r + r2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.4166666667r + 1 + r2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.4166666667r + r2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.4166666667r + r2 = 0 + 1
0.4166666667r + r2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.4166666667r + r2 = 1

The r term is 0.4166666667r.  Take half its coefficient (0.2083333334).
Square it (0.04340277781) and add it to both sides.

Add '0.04340277781' to each side of the equation.
0.4166666667r + 0.04340277781 + r2 = 1 + 0.04340277781

Reorder the terms:
0.04340277781 + 0.4166666667r + r2 = 1 + 0.04340277781

Combine like terms: 1 + 0.04340277781 = 1.04340277781
0.04340277781 + 0.4166666667r + r2 = 1.04340277781

Factor a perfect square on the left side:
(r + 0.2083333334)(r + 0.2083333334) = 1.04340277781

Calculate the square root of the right side: 1.021470889

Break this problem into two subproblems by setting 
(r + 0.2083333334) equal to 1.021470889 and -1.021470889.

Subproblem 1

r + 0.2083333334 = 1.021470889 Simplifying r + 0.2083333334 = 1.021470889 Reorder the terms: 0.2083333334 + r = 1.021470889 Solving 0.2083333334 + r = 1.021470889 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.2083333334' to each side of the equation. 0.2083333334 + -0.2083333334 + r = 1.021470889 + -0.2083333334 Combine like terms: 0.2083333334 + -0.2083333334 = 0.0000000000 0.0000000000 + r = 1.021470889 + -0.2083333334 r = 1.021470889 + -0.2083333334 Combine like terms: 1.021470889 + -0.2083333334 = 0.8131375556 r = 0.8131375556 Simplifying r = 0.8131375556

Subproblem 2

r + 0.2083333334 = -1.021470889 Simplifying r + 0.2083333334 = -1.021470889 Reorder the terms: 0.2083333334 + r = -1.021470889 Solving 0.2083333334 + r = -1.021470889 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.2083333334' to each side of the equation. 0.2083333334 + -0.2083333334 + r = -1.021470889 + -0.2083333334 Combine like terms: 0.2083333334 + -0.2083333334 = 0.0000000000 0.0000000000 + r = -1.021470889 + -0.2083333334 r = -1.021470889 + -0.2083333334 Combine like terms: -1.021470889 + -0.2083333334 = -1.2298042224 r = -1.2298042224 Simplifying r = -1.2298042224

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.8131375556, -1.2298042224}

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