3*x^2-16*x+15=0

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Solution for 3*x^2-16*x+15=0 equation:


Simplifying
3x2 + -16x + 15 = 0

Reorder the terms:
15 + -16x + 3x2 = 0

Solving
15 + -16x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
5 + -5.333333333x + x2 = 0

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + -5.333333333x + -5 + x2 = 0 + -5

Reorder the terms:
5 + -5 + -5.333333333x + x2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -5.333333333x + x2 = 0 + -5
-5.333333333x + x2 = 0 + -5

Combine like terms: 0 + -5 = -5
-5.333333333x + x2 = -5

The x term is -5.333333333x.  Take half its coefficient (-2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
-5.333333333x + 7.111111113 + x2 = -5 + 7.111111113

Reorder the terms:
7.111111113 + -5.333333333x + x2 = -5 + 7.111111113

Combine like terms: -5 + 7.111111113 = 2.111111113
7.111111113 + -5.333333333x + x2 = 2.111111113

Factor a perfect square on the left side:
(x + -2.666666667)(x + -2.666666667) = 2.111111113

Calculate the square root of the right side: 1.452966315

Break this problem into two subproblems by setting 
(x + -2.666666667) equal to 1.452966315 and -1.452966315.

Subproblem 1

x + -2.666666667 = 1.452966315 Simplifying x + -2.666666667 = 1.452966315 Reorder the terms: -2.666666667 + x = 1.452966315 Solving -2.666666667 + x = 1.452966315 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x = 1.452966315 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x = 1.452966315 + 2.666666667 x = 1.452966315 + 2.666666667 Combine like terms: 1.452966315 + 2.666666667 = 4.119632982 x = 4.119632982 Simplifying x = 4.119632982

Subproblem 2

x + -2.666666667 = -1.452966315 Simplifying x + -2.666666667 = -1.452966315 Reorder the terms: -2.666666667 + x = -1.452966315 Solving -2.666666667 + x = -1.452966315 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x = -1.452966315 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x = -1.452966315 + 2.666666667 x = -1.452966315 + 2.666666667 Combine like terms: -1.452966315 + 2.666666667 = 1.213700352 x = 1.213700352 Simplifying x = 1.213700352

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.119632982, 1.213700352}

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