0=d^2-20d+76

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Solution for 0=d^2-20d+76 equation:


Simplifying
0 = d2 + -20d + 76

Reorder the terms:
0 = 76 + -20d + d2

Solving
0 = 76 + -20d + d2

Solving for variable 'd'.

Combine like terms: 0 + -76 = -76
-76 + 20d + -1d2 = 76 + -20d + d2 + -76 + 20d + -1d2

Reorder the terms:
-76 + 20d + -1d2 = 76 + -76 + -20d + 20d + d2 + -1d2

Combine like terms: 76 + -76 = 0
-76 + 20d + -1d2 = 0 + -20d + 20d + d2 + -1d2
-76 + 20d + -1d2 = -20d + 20d + d2 + -1d2

Combine like terms: -20d + 20d = 0
-76 + 20d + -1d2 = 0 + d2 + -1d2
-76 + 20d + -1d2 = d2 + -1d2

Combine like terms: d2 + -1d2 = 0
-76 + 20d + -1d2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
76 + -20d + d2 = 0

Move the constant term to the right:

Add '-76' to each side of the equation.
76 + -20d + -76 + d2 = 0 + -76

Reorder the terms:
76 + -76 + -20d + d2 = 0 + -76

Combine like terms: 76 + -76 = 0
0 + -20d + d2 = 0 + -76
-20d + d2 = 0 + -76

Combine like terms: 0 + -76 = -76
-20d + d2 = -76

The d term is -20d.  Take half its coefficient (-10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
-20d + 100 + d2 = -76 + 100

Reorder the terms:
100 + -20d + d2 = -76 + 100

Combine like terms: -76 + 100 = 24
100 + -20d + d2 = 24

Factor a perfect square on the left side:
(d + -10)(d + -10) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(d + -10) equal to 4.898979486 and -4.898979486.

Subproblem 1

d + -10 = 4.898979486 Simplifying d + -10 = 4.898979486 Reorder the terms: -10 + d = 4.898979486 Solving -10 + d = 4.898979486 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + d = 4.898979486 + 10 Combine like terms: -10 + 10 = 0 0 + d = 4.898979486 + 10 d = 4.898979486 + 10 Combine like terms: 4.898979486 + 10 = 14.898979486 d = 14.898979486 Simplifying d = 14.898979486

Subproblem 2

d + -10 = -4.898979486 Simplifying d + -10 = -4.898979486 Reorder the terms: -10 + d = -4.898979486 Solving -10 + d = -4.898979486 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + d = -4.898979486 + 10 Combine like terms: -10 + 10 = 0 0 + d = -4.898979486 + 10 d = -4.898979486 + 10 Combine like terms: -4.898979486 + 10 = 5.101020514 d = 5.101020514 Simplifying d = 5.101020514

Solution

The solution to the problem is based on the solutions from the subproblems. d = {14.898979486, 5.101020514}

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