3x^2=7x+8

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Solution for 3x^2=7x+8 equation: 


Simplifying
3x2 = 7x + 8

Reorder the terms:
3x2 = 8 + 7x

Solving
3x2 = 8 + 7x

Solving for variable 'x'.

Reorder the terms:
-8 + -7x + 3x2 = 8 + 7x + -8 + -7x

Reorder the terms:
-8 + -7x + 3x2 = 8 + -8 + 7x + -7x

Combine like terms: 8 + -8 = 0
-8 + -7x + 3x2 = 0 + 7x + -7x
-8 + -7x + 3x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
-8 + -7x + 3x2 = 0

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

Divide each side by '3'.
-2.666666667 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + -2.333333333x + 2.666666667 + x2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + -2.333333333x + x2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + -2.333333333x + x2 = 0 + 2.666666667
-2.333333333x + x2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
-2.333333333x + x2 = 2.666666667

The x term is -2.333333333x.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333x + 1.361111112 + x2 = 2.666666667 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = 2.666666667 + 1.361111112

Combine like terms: 2.666666667 + 1.361111112 = 4.027777779
1.361111112 + -2.333333333x + x2 = 4.027777779

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 4.027777779

Calculate the square root of the right side: 2.00693243

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 2.00693243 and -2.00693243.


Subproblem 1
x + -1.166666667 = 2.00693243

Simplifying
x + -1.166666667 = 2.00693243

Reorder the terms:
-1.166666667 + x = 2.00693243

Solving
-1.166666667 + x = 2.00693243

Solving for variable 'x'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = 2.00693243 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = 2.00693243 + 1.166666667
x = 2.00693243 + 1.166666667

Combine like terms: 2.00693243 + 1.166666667 = 3.173599097
x = 3.173599097

Simplifying
x = 3.173599097


Subproblem 2
x + -1.166666667 = -2.00693243

Simplifying
x + -1.166666667 = -2.00693243

Reorder the terms:
-1.166666667 + x = -2.00693243

Solving
-1.166666667 + x = -2.00693243

Solving for variable 'x'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = -2.00693243 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = -2.00693243 + 1.166666667
x = -2.00693243 + 1.166666667

Combine like terms: -2.00693243 + 1.166666667 = -0.840265763
x = -0.840265763

Simplifying
x = -0.840265763


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.173599097, -0.840265763}

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