5x(3x+7)=83

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Solution for 5x(3x+7)=83 equation:


Simplifying
5x(3x + 7) = 83

Reorder the terms:
5x(7 + 3x) = 83
(7 * 5x + 3x * 5x) = 83
(35x + 15x2) = 83

Solving
35x + 15x2 = 83

Solving for variable 'x'.

Reorder the terms:
-83 + 35x + 15x2 = 83 + -83

Combine like terms: 83 + -83 = 0
-83 + 35x + 15x2 = 0

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

Divide each side by '15'.
-5.533333333 + 2.333333333x + x2 = 0

Move the constant term to the right:

Add '5.533333333' to each side of the equation.
-5.533333333 + 2.333333333x + 5.533333333 + x2 = 0 + 5.533333333

Reorder the terms:
-5.533333333 + 5.533333333 + 2.333333333x + x2 = 0 + 5.533333333

Combine like terms: -5.533333333 + 5.533333333 = 0.000000000
0.000000000 + 2.333333333x + x2 = 0 + 5.533333333
2.333333333x + x2 = 0 + 5.533333333

Combine like terms: 0 + 5.533333333 = 5.533333333
2.333333333x + x2 = 5.533333333

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 = 5.533333333 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333x + x2 = 5.533333333 + 1.361111112

Combine like terms: 5.533333333 + 1.361111112 = 6.894444445
1.361111112 + 2.333333333x + x2 = 6.894444445

Factor a perfect square on the left side:
(x + 1.166666667)(x + 1.166666667) = 6.894444445

Calculate the square root of the right side: 2.625727413

Break this problem into two subproblems by setting 
(x + 1.166666667) equal to 2.625727413 and -2.625727413.

Subproblem 1

x + 1.166666667 = 2.625727413 Simplifying x + 1.166666667 = 2.625727413 Reorder the terms: 1.166666667 + x = 2.625727413 Solving 1.166666667 + x = 2.625727413 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 = 2.625727413 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 2.625727413 + -1.166666667 x = 2.625727413 + -1.166666667 Combine like terms: 2.625727413 + -1.166666667 = 1.459060746 x = 1.459060746 Simplifying x = 1.459060746

Subproblem 2

x + 1.166666667 = -2.625727413 Simplifying x + 1.166666667 = -2.625727413 Reorder the terms: 1.166666667 + x = -2.625727413 Solving 1.166666667 + x = -2.625727413 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 = -2.625727413 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -2.625727413 + -1.166666667 x = -2.625727413 + -1.166666667 Combine like terms: -2.625727413 + -1.166666667 = -3.79239408 x = -3.79239408 Simplifying x = -3.79239408

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.459060746, -3.79239408}

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