9r^2-5r-10=0

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Solution for 9r^2-5r-10=0 equation:


Simplifying
9r2 + -5r + -10 = 0

Reorder the terms:
-10 + -5r + 9r2 = 0

Solving
-10 + -5r + 9r2 = 0

Solving for variable 'r'.

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

Divide each side by '9'.
-1.111111111 + -0.5555555556r + r2 = 0

Move the constant term to the right:

Add '1.111111111' to each side of the equation.
-1.111111111 + -0.5555555556r + 1.111111111 + r2 = 0 + 1.111111111

Reorder the terms:
-1.111111111 + 1.111111111 + -0.5555555556r + r2 = 0 + 1.111111111

Combine like terms: -1.111111111 + 1.111111111 = 0.000000000
0.000000000 + -0.5555555556r + r2 = 0 + 1.111111111
-0.5555555556r + r2 = 0 + 1.111111111

Combine like terms: 0 + 1.111111111 = 1.111111111
-0.5555555556r + r2 = 1.111111111

The r term is -0.5555555556r.  Take half its coefficient (-0.2777777778).
Square it (0.07716049384) and add it to both sides.

Add '0.07716049384' to each side of the equation.
-0.5555555556r + 0.07716049384 + r2 = 1.111111111 + 0.07716049384

Reorder the terms:
0.07716049384 + -0.5555555556r + r2 = 1.111111111 + 0.07716049384

Combine like terms: 1.111111111 + 0.07716049384 = 1.18827160484
0.07716049384 + -0.5555555556r + r2 = 1.18827160484

Factor a perfect square on the left side:
(r + -0.2777777778)(r + -0.2777777778) = 1.18827160484

Calculate the square root of the right side: 1.090078715

Break this problem into two subproblems by setting 
(r + -0.2777777778) equal to 1.090078715 and -1.090078715.

Subproblem 1

r + -0.2777777778 = 1.090078715 Simplifying r + -0.2777777778 = 1.090078715 Reorder the terms: -0.2777777778 + r = 1.090078715 Solving -0.2777777778 + r = 1.090078715 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + r = 1.090078715 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + r = 1.090078715 + 0.2777777778 r = 1.090078715 + 0.2777777778 Combine like terms: 1.090078715 + 0.2777777778 = 1.3678564928 r = 1.3678564928 Simplifying r = 1.3678564928

Subproblem 2

r + -0.2777777778 = -1.090078715 Simplifying r + -0.2777777778 = -1.090078715 Reorder the terms: -0.2777777778 + r = -1.090078715 Solving -0.2777777778 + r = -1.090078715 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + r = -1.090078715 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + r = -1.090078715 + 0.2777777778 r = -1.090078715 + 0.2777777778 Combine like terms: -1.090078715 + 0.2777777778 = -0.8123009372 r = -0.8123009372 Simplifying r = -0.8123009372

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.3678564928, -0.8123009372}

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