3d(d+5)=23

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Solution for 3d(d+5)=23 equation:


Simplifying
3d(d + 5) = 23

Reorder the terms:
3d(5 + d) = 23
(5 * 3d + d * 3d) = 23
(15d + 3d2) = 23

Solving
15d + 3d2 = 23

Solving for variable 'd'.

Reorder the terms:
-23 + 15d + 3d2 = 23 + -23

Combine like terms: 23 + -23 = 0
-23 + 15d + 3d2 = 0

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

Divide each side by '3'.
-7.666666667 + 5d + d2 = 0

Move the constant term to the right:

Add '7.666666667' to each side of the equation.
-7.666666667 + 5d + 7.666666667 + d2 = 0 + 7.666666667

Reorder the terms:
-7.666666667 + 7.666666667 + 5d + d2 = 0 + 7.666666667

Combine like terms: -7.666666667 + 7.666666667 = 0.000000000
0.000000000 + 5d + d2 = 0 + 7.666666667
5d + d2 = 0 + 7.666666667

Combine like terms: 0 + 7.666666667 = 7.666666667
5d + d2 = 7.666666667

The d term is 5d.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5d + 6.25 + d2 = 7.666666667 + 6.25

Reorder the terms:
6.25 + 5d + d2 = 7.666666667 + 6.25

Combine like terms: 7.666666667 + 6.25 = 13.916666667
6.25 + 5d + d2 = 13.916666667

Factor a perfect square on the left side:
(d + 2.5)(d + 2.5) = 13.916666667

Calculate the square root of the right side: 3.730504881

Break this problem into two subproblems by setting 
(d + 2.5) equal to 3.730504881 and -3.730504881.

Subproblem 1

d + 2.5 = 3.730504881 Simplifying d + 2.5 = 3.730504881 Reorder the terms: 2.5 + d = 3.730504881 Solving 2.5 + d = 3.730504881 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + d = 3.730504881 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + d = 3.730504881 + -2.5 d = 3.730504881 + -2.5 Combine like terms: 3.730504881 + -2.5 = 1.230504881 d = 1.230504881 Simplifying d = 1.230504881

Subproblem 2

d + 2.5 = -3.730504881 Simplifying d + 2.5 = -3.730504881 Reorder the terms: 2.5 + d = -3.730504881 Solving 2.5 + d = -3.730504881 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + d = -3.730504881 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + d = -3.730504881 + -2.5 d = -3.730504881 + -2.5 Combine like terms: -3.730504881 + -2.5 = -6.230504881 d = -6.230504881 Simplifying d = -6.230504881

Solution

The solution to the problem is based on the solutions from the subproblems. d = {1.230504881, -6.230504881}

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