x(3x+5)=105

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Solution for x(3x+5)=105 equation:


Simplifying
x(3x + 5) = 105

Reorder the terms:
x(5 + 3x) = 105
(5 * x + 3x * x) = 105
(5x + 3x2) = 105

Solving
5x + 3x2 = 105

Solving for variable 'x'.

Reorder the terms:
-105 + 5x + 3x2 = 105 + -105

Combine like terms: 105 + -105 = 0
-105 + 5x + 3x2 = 0

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

Divide each side by '3'.
-35 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '35' to each side of the equation.
-35 + 1.666666667x + 35 + x2 = 0 + 35

Reorder the terms:
-35 + 35 + 1.666666667x + x2 = 0 + 35

Combine like terms: -35 + 35 = 0
0 + 1.666666667x + x2 = 0 + 35
1.666666667x + x2 = 0 + 35

Combine like terms: 0 + 35 = 35
1.666666667x + x2 = 35

The x term is 1.666666667x.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667x + 0.6944444447 + x2 = 35 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 35 + 0.6944444447

Combine like terms: 35 + 0.6944444447 = 35.6944444447
0.6944444447 + 1.666666667x + x2 = 35.6944444447

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 35.6944444447

Calculate the square root of the right side: 5.974482776

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 5.974482776 and -5.974482776.

Subproblem 1

x + 0.8333333335 = 5.974482776 Simplifying x + 0.8333333335 = 5.974482776 Reorder the terms: 0.8333333335 + x = 5.974482776 Solving 0.8333333335 + x = 5.974482776 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + x = 5.974482776 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = 5.974482776 + -0.8333333335 x = 5.974482776 + -0.8333333335 Combine like terms: 5.974482776 + -0.8333333335 = 5.1411494425 x = 5.1411494425 Simplifying x = 5.1411494425

Subproblem 2

x + 0.8333333335 = -5.974482776 Simplifying x + 0.8333333335 = -5.974482776 Reorder the terms: 0.8333333335 + x = -5.974482776 Solving 0.8333333335 + x = -5.974482776 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + x = -5.974482776 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = -5.974482776 + -0.8333333335 x = -5.974482776 + -0.8333333335 Combine like terms: -5.974482776 + -0.8333333335 = -6.8078161095 x = -6.8078161095 Simplifying x = -6.8078161095

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.1411494425, -6.8078161095}

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