6.5x^2-22.5x+15=0

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Solution for 6.5x^2-22.5x+15=0 equation:


Simplifying
6.5x2 + -22.5x + 15 = 0

Reorder the terms:
15 + -22.5x + 6.5x2 = 0

Solving
15 + -22.5x + 6.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '6.5'.
2.307692308 + -3.461538462x + x2 = 0

Move the constant term to the right:

Add '-2.307692308' to each side of the equation.
2.307692308 + -3.461538462x + -2.307692308 + x2 = 0 + -2.307692308

Reorder the terms:
2.307692308 + -2.307692308 + -3.461538462x + x2 = 0 + -2.307692308

Combine like terms: 2.307692308 + -2.307692308 = 0.000000000
0.000000000 + -3.461538462x + x2 = 0 + -2.307692308
-3.461538462x + x2 = 0 + -2.307692308

Combine like terms: 0 + -2.307692308 = -2.307692308
-3.461538462x + x2 = -2.307692308

The x term is -3.461538462x.  Take half its coefficient (-1.730769231).
Square it (2.995562131) and add it to both sides.

Add '2.995562131' to each side of the equation.
-3.461538462x + 2.995562131 + x2 = -2.307692308 + 2.995562131

Reorder the terms:
2.995562131 + -3.461538462x + x2 = -2.307692308 + 2.995562131

Combine like terms: -2.307692308 + 2.995562131 = 0.687869823
2.995562131 + -3.461538462x + x2 = 0.687869823

Factor a perfect square on the left side:
(x + -1.730769231)(x + -1.730769231) = 0.687869823

Calculate the square root of the right side: 0.829379179

Break this problem into two subproblems by setting 
(x + -1.730769231) equal to 0.829379179 and -0.829379179.

Subproblem 1

x + -1.730769231 = 0.829379179 Simplifying x + -1.730769231 = 0.829379179 Reorder the terms: -1.730769231 + x = 0.829379179 Solving -1.730769231 + x = 0.829379179 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.730769231' to each side of the equation. -1.730769231 + 1.730769231 + x = 0.829379179 + 1.730769231 Combine like terms: -1.730769231 + 1.730769231 = 0.000000000 0.000000000 + x = 0.829379179 + 1.730769231 x = 0.829379179 + 1.730769231 Combine like terms: 0.829379179 + 1.730769231 = 2.56014841 x = 2.56014841 Simplifying x = 2.56014841

Subproblem 2

x + -1.730769231 = -0.829379179 Simplifying x + -1.730769231 = -0.829379179 Reorder the terms: -1.730769231 + x = -0.829379179 Solving -1.730769231 + x = -0.829379179 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.730769231' to each side of the equation. -1.730769231 + 1.730769231 + x = -0.829379179 + 1.730769231 Combine like terms: -1.730769231 + 1.730769231 = 0.000000000 0.000000000 + x = -0.829379179 + 1.730769231 x = -0.829379179 + 1.730769231 Combine like terms: -0.829379179 + 1.730769231 = 0.901390052 x = 0.901390052 Simplifying x = 0.901390052

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.56014841, 0.901390052}

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