3d^2=-5d+19

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Solution for 3d^2=-5d+19 equation:


Simplifying
3d2 = -5d + 19

Reorder the terms:
3d2 = 19 + -5d

Solving
3d2 = 19 + -5d

Solving for variable 'd'.

Reorder the terms:
-19 + 5d + 3d2 = 19 + -5d + -19 + 5d

Reorder the terms:
-19 + 5d + 3d2 = 19 + -19 + -5d + 5d

Combine like terms: 19 + -19 = 0
-19 + 5d + 3d2 = 0 + -5d + 5d
-19 + 5d + 3d2 = -5d + 5d

Combine like terms: -5d + 5d = 0
-19 + 5d + 3d2 = 0

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

Divide each side by '3'.
-6.333333333 + 1.666666667d + d2 = 0

Move the constant term to the right:

Add '6.333333333' to each side of the equation.
-6.333333333 + 1.666666667d + 6.333333333 + d2 = 0 + 6.333333333

Reorder the terms:
-6.333333333 + 6.333333333 + 1.666666667d + d2 = 0 + 6.333333333

Combine like terms: -6.333333333 + 6.333333333 = 0.000000000
0.000000000 + 1.666666667d + d2 = 0 + 6.333333333
1.666666667d + d2 = 0 + 6.333333333

Combine like terms: 0 + 6.333333333 = 6.333333333
1.666666667d + d2 = 6.333333333

The d term is 1.666666667d.  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.
1.666666667d + 0.6944444447 + d2 = 6.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667d + d2 = 6.333333333 + 0.6944444447

Combine like terms: 6.333333333 + 0.6944444447 = 7.0277777777
0.6944444447 + 1.666666667d + d2 = 7.0277777777

Factor a perfect square on the left side:
(d + 0.8333333335)(d + 0.8333333335) = 7.0277777777

Calculate the square root of the right side: 2.65099562

Break this problem into two subproblems by setting 
(d + 0.8333333335) equal to 2.65099562 and -2.65099562.

Subproblem 1

d + 0.8333333335 = 2.65099562 Simplifying d + 0.8333333335 = 2.65099562 Reorder the terms: 0.8333333335 + d = 2.65099562 Solving 0.8333333335 + d = 2.65099562 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + d = 2.65099562 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + d = 2.65099562 + -0.8333333335 d = 2.65099562 + -0.8333333335 Combine like terms: 2.65099562 + -0.8333333335 = 1.8176622865 d = 1.8176622865 Simplifying d = 1.8176622865

Subproblem 2

d + 0.8333333335 = -2.65099562 Simplifying d + 0.8333333335 = -2.65099562 Reorder the terms: 0.8333333335 + d = -2.65099562 Solving 0.8333333335 + d = -2.65099562 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + d = -2.65099562 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + d = -2.65099562 + -0.8333333335 d = -2.65099562 + -0.8333333335 Combine like terms: -2.65099562 + -0.8333333335 = -3.4843289535 d = -3.4843289535 Simplifying d = -3.4843289535

Solution

The solution to the problem is based on the solutions from the subproblems. d = {1.8176622865, -3.4843289535}

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