3x(5+3x)=10

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Solution for 3x(5+3x)=10 equation:


Simplifying
3x(5 + 3x) = 10
(5 * 3x + 3x * 3x) = 10
(15x + 9x2) = 10

Solving
15x + 9x2 = 10

Solving for variable 'x'.

Reorder the terms:
-10 + 15x + 9x2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 15x + 9x2 = 0

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

Divide each side by '9'.
-1.111111111 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '1.111111111' to each side of the equation.
-1.111111111 + 1.666666667x + 1.111111111 + x2 = 0 + 1.111111111

Reorder the terms:
-1.111111111 + 1.111111111 + 1.666666667x + x2 = 0 + 1.111111111

Combine like terms: -1.111111111 + 1.111111111 = 0.000000000
0.000000000 + 1.666666667x + x2 = 0 + 1.111111111
1.666666667x + x2 = 0 + 1.111111111

Combine like terms: 0 + 1.111111111 = 1.111111111
1.666666667x + x2 = 1.111111111

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 = 1.111111111 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 1.111111111 + 0.6944444447

Combine like terms: 1.111111111 + 0.6944444447 = 1.8055555557
0.6944444447 + 1.666666667x + x2 = 1.8055555557

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 1.8055555557

Calculate the square root of the right side: 1.343709625

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 1.343709625 and -1.343709625.

Subproblem 1

x + 0.8333333335 = 1.343709625 Simplifying x + 0.8333333335 = 1.343709625 Reorder the terms: 0.8333333335 + x = 1.343709625 Solving 0.8333333335 + x = 1.343709625 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 = 1.343709625 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = 1.343709625 + -0.8333333335 x = 1.343709625 + -0.8333333335 Combine like terms: 1.343709625 + -0.8333333335 = 0.5103762915 x = 0.5103762915 Simplifying x = 0.5103762915

Subproblem 2

x + 0.8333333335 = -1.343709625 Simplifying x + 0.8333333335 = -1.343709625 Reorder the terms: 0.8333333335 + x = -1.343709625 Solving 0.8333333335 + x = -1.343709625 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 = -1.343709625 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = -1.343709625 + -0.8333333335 x = -1.343709625 + -0.8333333335 Combine like terms: -1.343709625 + -0.8333333335 = -2.1770429585 x = -2.1770429585 Simplifying x = -2.1770429585

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.5103762915, -2.1770429585}

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