15x^2+61x-20=0

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Solution for 15x^2+61x-20=0 equation:


Simplifying
15x2 + 61x + -20 = 0

Reorder the terms:
-20 + 61x + 15x2 = 0

Solving
-20 + 61x + 15x2 = 0

Solving for variable 'x'.

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

Divide each side by '15'.
-1.333333333 + 4.066666667x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + 4.066666667x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + 4.066666667x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + 4.066666667x + x2 = 0 + 1.333333333
4.066666667x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
4.066666667x + x2 = 1.333333333

The x term is 4.066666667x.  Take half its coefficient (2.033333334).
Square it (4.134444447) and add it to both sides.

Add '4.134444447' to each side of the equation.
4.066666667x + 4.134444447 + x2 = 1.333333333 + 4.134444447

Reorder the terms:
4.134444447 + 4.066666667x + x2 = 1.333333333 + 4.134444447

Combine like terms: 1.333333333 + 4.134444447 = 5.46777778
4.134444447 + 4.066666667x + x2 = 5.46777778

Factor a perfect square on the left side:
(x + 2.033333334)(x + 2.033333334) = 5.46777778

Calculate the square root of the right side: 2.338327988

Break this problem into two subproblems by setting 
(x + 2.033333334) equal to 2.338327988 and -2.338327988.

Subproblem 1

x + 2.033333334 = 2.338327988 Simplifying x + 2.033333334 = 2.338327988 Reorder the terms: 2.033333334 + x = 2.338327988 Solving 2.033333334 + x = 2.338327988 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.033333334' to each side of the equation. 2.033333334 + -2.033333334 + x = 2.338327988 + -2.033333334 Combine like terms: 2.033333334 + -2.033333334 = 0.000000000 0.000000000 + x = 2.338327988 + -2.033333334 x = 2.338327988 + -2.033333334 Combine like terms: 2.338327988 + -2.033333334 = 0.304994654 x = 0.304994654 Simplifying x = 0.304994654

Subproblem 2

x + 2.033333334 = -2.338327988 Simplifying x + 2.033333334 = -2.338327988 Reorder the terms: 2.033333334 + x = -2.338327988 Solving 2.033333334 + x = -2.338327988 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.033333334' to each side of the equation. 2.033333334 + -2.033333334 + x = -2.338327988 + -2.033333334 Combine like terms: 2.033333334 + -2.033333334 = 0.000000000 0.000000000 + x = -2.338327988 + -2.033333334 x = -2.338327988 + -2.033333334 Combine like terms: -2.338327988 + -2.033333334 = -4.371661322 x = -4.371661322 Simplifying x = -4.371661322

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.304994654, -4.371661322}

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