3x^2-7x-23=7

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Solution for 3x^2-7x-23=7 equation:


Simplifying
3x2 + -7x + -23 = 7

Reorder the terms:
-23 + -7x + 3x2 = 7

Solving
-23 + -7x + 3x2 = 7

Solving for variable 'x'.

Reorder the terms:
-23 + -7 + -7x + 3x2 = 7 + -7

Combine like terms: -23 + -7 = -30
-30 + -7x + 3x2 = 7 + -7

Combine like terms: 7 + -7 = 0
-30 + -7x + 3x2 = 0

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

Divide each side by '3'.
-10 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -2.333333333x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + -2.333333333x + x2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -2.333333333x + x2 = 0 + 10
-2.333333333x + x2 = 0 + 10

Combine like terms: 0 + 10 = 10
-2.333333333x + x2 = 10

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 = 10 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = 10 + 1.361111112

Combine like terms: 10 + 1.361111112 = 11.361111112
1.361111112 + -2.333333333x + x2 = 11.361111112

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 11.361111112

Calculate the square root of the right side: 3.370624736

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 3.370624736 and -3.370624736.

Subproblem 1

x + -1.166666667 = 3.370624736 Simplifying x + -1.166666667 = 3.370624736 Reorder the terms: -1.166666667 + x = 3.370624736 Solving -1.166666667 + x = 3.370624736 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 = 3.370624736 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = 3.370624736 + 1.166666667 x = 3.370624736 + 1.166666667 Combine like terms: 3.370624736 + 1.166666667 = 4.537291403 x = 4.537291403 Simplifying x = 4.537291403

Subproblem 2

x + -1.166666667 = -3.370624736 Simplifying x + -1.166666667 = -3.370624736 Reorder the terms: -1.166666667 + x = -3.370624736 Solving -1.166666667 + x = -3.370624736 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 = -3.370624736 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = -3.370624736 + 1.166666667 x = -3.370624736 + 1.166666667 Combine like terms: -3.370624736 + 1.166666667 = -2.203958069 x = -2.203958069 Simplifying x = -2.203958069

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.537291403, -2.203958069}

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