3(x^2-4)+7(x-3)=6

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Solution for 3(x^2-4)+7(x-3)=6 equation:


Simplifying
3(x2 + -4) + 7(x + -3) = 6

Reorder the terms:
3(-4 + x2) + 7(x + -3) = 6
(-4 * 3 + x2 * 3) + 7(x + -3) = 6
(-12 + 3x2) + 7(x + -3) = 6

Reorder the terms:
-12 + 3x2 + 7(-3 + x) = 6
-12 + 3x2 + (-3 * 7 + x * 7) = 6
-12 + 3x2 + (-21 + 7x) = 6

Reorder the terms:
-12 + -21 + 7x + 3x2 = 6

Combine like terms: -12 + -21 = -33
-33 + 7x + 3x2 = 6

Solving
-33 + 7x + 3x2 = 6

Solving for variable 'x'.

Reorder the terms:
-33 + -6 + 7x + 3x2 = 6 + -6

Combine like terms: -33 + -6 = -39
-39 + 7x + 3x2 = 6 + -6

Combine like terms: 6 + -6 = 0
-39 + 7x + 3x2 = 0

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

Divide each side by '3'.
-13 + 2.333333333x + x2 = 0

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + 2.333333333x + 13 + x2 = 0 + 13

Reorder the terms:
-13 + 13 + 2.333333333x + x2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + 2.333333333x + x2 = 0 + 13
2.333333333x + x2 = 0 + 13

Combine like terms: 0 + 13 = 13
2.333333333x + x2 = 13

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 = 13 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333x + x2 = 13 + 1.361111112

Combine like terms: 13 + 1.361111112 = 14.361111112
1.361111112 + 2.333333333x + x2 = 14.361111112

Factor a perfect square on the left side:
(x + 1.166666667)(x + 1.166666667) = 14.361111112

Calculate the square root of the right side: 3.789605667

Break this problem into two subproblems by setting 
(x + 1.166666667) equal to 3.789605667 and -3.789605667.

Subproblem 1

x + 1.166666667 = 3.789605667 Simplifying x + 1.166666667 = 3.789605667 Reorder the terms: 1.166666667 + x = 3.789605667 Solving 1.166666667 + x = 3.789605667 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 = 3.789605667 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 3.789605667 + -1.166666667 x = 3.789605667 + -1.166666667 Combine like terms: 3.789605667 + -1.166666667 = 2.622939 x = 2.622939 Simplifying x = 2.622939

Subproblem 2

x + 1.166666667 = -3.789605667 Simplifying x + 1.166666667 = -3.789605667 Reorder the terms: 1.166666667 + x = -3.789605667 Solving 1.166666667 + x = -3.789605667 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 = -3.789605667 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -3.789605667 + -1.166666667 x = -3.789605667 + -1.166666667 Combine like terms: -3.789605667 + -1.166666667 = -4.956272334 x = -4.956272334 Simplifying x = -4.956272334

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.622939, -4.956272334}

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