4r^2+10r-1=0

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Solution for 4r^2+10r-1=0 equation:


Simplifying
4r2 + 10r + -1 = 0

Reorder the terms:
-1 + 10r + 4r2 = 0

Solving
-1 + 10r + 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'.
-0.25 + 2.5r + r2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 2.5r + 0.25 + r2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 2.5r + r2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 2.5r + r2 = 0 + 0.25
2.5r + r2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
2.5r + r2 = 0.25

The r term is 2.5r.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5r + 1.5625 + r2 = 0.25 + 1.5625

Reorder the terms:
1.5625 + 2.5r + r2 = 0.25 + 1.5625

Combine like terms: 0.25 + 1.5625 = 1.8125
1.5625 + 2.5r + r2 = 1.8125

Factor a perfect square on the left side:
(r + 1.25)(r + 1.25) = 1.8125

Calculate the square root of the right side: 1.346291202

Break this problem into two subproblems by setting 
(r + 1.25) equal to 1.346291202 and -1.346291202.

Subproblem 1

r + 1.25 = 1.346291202 Simplifying r + 1.25 = 1.346291202 Reorder the terms: 1.25 + r = 1.346291202 Solving 1.25 + r = 1.346291202 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + r = 1.346291202 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + r = 1.346291202 + -1.25 r = 1.346291202 + -1.25 Combine like terms: 1.346291202 + -1.25 = 0.096291202 r = 0.096291202 Simplifying r = 0.096291202

Subproblem 2

r + 1.25 = -1.346291202 Simplifying r + 1.25 = -1.346291202 Reorder the terms: 1.25 + r = -1.346291202 Solving 1.25 + r = -1.346291202 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + r = -1.346291202 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + r = -1.346291202 + -1.25 r = -1.346291202 + -1.25 Combine like terms: -1.346291202 + -1.25 = -2.596291202 r = -2.596291202 Simplifying r = -2.596291202

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.096291202, -2.596291202}

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