10x^2+2x+1=16x^2-24x-9

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Solution for 10x^2+2x+1=16x^2-24x-9 equation: 


Simplifying
10x2 + 2x + 1 = 16x2 + -24x + -9

Reorder the terms:
1 + 2x + 10x2 = 16x2 + -24x + -9

Reorder the terms:
1 + 2x + 10x2 = -9 + -24x + 16x2

Solving
1 + 2x + 10x2 = -9 + -24x + 16x2

Solving for variable 'x'.

Reorder the terms:
1 + 9 + 2x + 24x + 10x2 + -16x2 = -9 + -24x + 16x2 + 9 + 24x + -16x2

Combine like terms: 1 + 9 = 10
10 + 2x + 24x + 10x2 + -16x2 = -9 + -24x + 16x2 + 9 + 24x + -16x2

Combine like terms: 2x + 24x = 26x
10 + 26x + 10x2 + -16x2 = -9 + -24x + 16x2 + 9 + 24x + -16x2

Combine like terms: 10x2 + -16x2 = -6x2
10 + 26x + -6x2 = -9 + -24x + 16x2 + 9 + 24x + -16x2

Reorder the terms:
10 + 26x + -6x2 = -9 + 9 + -24x + 24x + 16x2 + -16x2

Combine like terms: -9 + 9 = 0
10 + 26x + -6x2 = 0 + -24x + 24x + 16x2 + -16x2
10 + 26x + -6x2 = -24x + 24x + 16x2 + -16x2

Combine like terms: -24x + 24x = 0
10 + 26x + -6x2 = 0 + 16x2 + -16x2
10 + 26x + -6x2 = 16x2 + -16x2

Combine like terms: 16x2 + -16x2 = 0
10 + 26x + -6x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(5 + 13x + -3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(5 + 13x + -3x2)' equal to zero and attempt to solve:

Simplifying
5 + 13x + -3x2 = 0

Solving
5 + 13x + -3x2 = 0

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

Divide each side by '-3'.
-1.666666667 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -4.333333333x + 1.666666667 + x2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -4.333333333x + x2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -4.333333333x + x2 = 0 + 1.666666667
-4.333333333x + x2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-4.333333333x + x2 = 1.666666667

The x term is -4.333333333x.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333x + 4.694444446 + x2 = 1.666666667 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = 1.666666667 + 4.694444446

Combine like terms: 1.666666667 + 4.694444446 = 6.361111113
4.694444446 + -4.333333333x + x2 = 6.361111113

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 6.361111113

Calculate the square root of the right side: 2.522124325

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 2.522124325 and -2.522124325.


Subproblem 1
x + -2.166666667 = 2.522124325

Simplifying
x + -2.166666667 = 2.522124325

Reorder the terms:
-2.166666667 + x = 2.522124325

Solving
-2.166666667 + x = 2.522124325

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = 2.522124325 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = 2.522124325 + 2.166666667
x = 2.522124325 + 2.166666667

Combine like terms: 2.522124325 + 2.166666667 = 4.688790992
x = 4.688790992

Simplifying
x = 4.688790992


Subproblem 2
x + -2.166666667 = -2.522124325

Simplifying
x + -2.166666667 = -2.522124325

Reorder the terms:
-2.166666667 + x = -2.522124325

Solving
-2.166666667 + x = -2.522124325

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = -2.522124325 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = -2.522124325 + 2.166666667
x = -2.522124325 + 2.166666667

Combine like terms: -2.522124325 + 2.166666667 = -0.355457658
x = -0.355457658

Simplifying
x = -0.355457658


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.688790992, -0.355457658}
Solution
x = {4.688790992, -0.355457658}

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