7-3x=3x^2-10x+4

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Solution for 7-3x=3x^2-10x+4 equation:


Simplifying
7 + -3x = 3x2 + -10x + 4

Reorder the terms:
7 + -3x = 4 + -10x + 3x2

Solving
7 + -3x = 4 + -10x + 3x2

Solving for variable 'x'.

Reorder the terms:
7 + -4 + -3x + 10x + -3x2 = 4 + -10x + 3x2 + -4 + 10x + -3x2

Combine like terms: 7 + -4 = 3
3 + -3x + 10x + -3x2 = 4 + -10x + 3x2 + -4 + 10x + -3x2

Combine like terms: -3x + 10x = 7x
3 + 7x + -3x2 = 4 + -10x + 3x2 + -4 + 10x + -3x2

Reorder the terms:
3 + 7x + -3x2 = 4 + -4 + -10x + 10x + 3x2 + -3x2

Combine like terms: 4 + -4 = 0
3 + 7x + -3x2 = 0 + -10x + 10x + 3x2 + -3x2
3 + 7x + -3x2 = -10x + 10x + 3x2 + -3x2

Combine like terms: -10x + 10x = 0
3 + 7x + -3x2 = 0 + 3x2 + -3x2
3 + 7x + -3x2 = 3x2 + -3x2

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

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

Divide each side by '-3'.
-1 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2.333333333x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -2.333333333x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2.333333333x + x2 = 0 + 1
-2.333333333x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2.333333333x + x2 = 1

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 = 1 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = 1 + 1.361111112

Combine like terms: 1 + 1.361111112 = 2.361111112
1.361111112 + -2.333333333x + x2 = 2.361111112

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 2.361111112

Calculate the square root of the right side: 1.536590743

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 1.536590743 and -1.536590743.

Subproblem 1

x + -1.166666667 = 1.536590743 Simplifying x + -1.166666667 = 1.536590743 Reorder the terms: -1.166666667 + x = 1.536590743 Solving -1.166666667 + x = 1.536590743 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 = 1.536590743 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = 1.536590743 + 1.166666667 x = 1.536590743 + 1.166666667 Combine like terms: 1.536590743 + 1.166666667 = 2.70325741 x = 2.70325741 Simplifying x = 2.70325741

Subproblem 2

x + -1.166666667 = -1.536590743 Simplifying x + -1.166666667 = -1.536590743 Reorder the terms: -1.166666667 + x = -1.536590743 Solving -1.166666667 + x = -1.536590743 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 = -1.536590743 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = -1.536590743 + 1.166666667 x = -1.536590743 + 1.166666667 Combine like terms: -1.536590743 + 1.166666667 = -0.369924076 x = -0.369924076 Simplifying x = -0.369924076

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.70325741, -0.369924076}

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