24t-t^2=72

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Solution for 24t-t^2=72 equation:


Simplifying
24t + -1t2 = 72

Solving
24t + -1t2 = 72

Solving for variable 't'.

Reorder the terms:
-72 + 24t + -1t2 = 72 + -72

Combine like terms: 72 + -72 = 0
-72 + 24t + -1t2 = 0

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

Divide each side by '-1'.
72 + -24t + t2 = 0

Move the constant term to the right:

Add '-72' to each side of the equation.
72 + -24t + -72 + t2 = 0 + -72

Reorder the terms:
72 + -72 + -24t + t2 = 0 + -72

Combine like terms: 72 + -72 = 0
0 + -24t + t2 = 0 + -72
-24t + t2 = 0 + -72

Combine like terms: 0 + -72 = -72
-24t + t2 = -72

The t term is -24t.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24t + 144 + t2 = -72 + 144

Reorder the terms:
144 + -24t + t2 = -72 + 144

Combine like terms: -72 + 144 = 72
144 + -24t + t2 = 72

Factor a perfect square on the left side:
(t + -12)(t + -12) = 72

Calculate the square root of the right side: 8.485281374

Break this problem into two subproblems by setting 
(t + -12) equal to 8.485281374 and -8.485281374.

Subproblem 1

t + -12 = 8.485281374 Simplifying t + -12 = 8.485281374 Reorder the terms: -12 + t = 8.485281374 Solving -12 + t = 8.485281374 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + t = 8.485281374 + 12 Combine like terms: -12 + 12 = 0 0 + t = 8.485281374 + 12 t = 8.485281374 + 12 Combine like terms: 8.485281374 + 12 = 20.485281374 t = 20.485281374 Simplifying t = 20.485281374

Subproblem 2

t + -12 = -8.485281374 Simplifying t + -12 = -8.485281374 Reorder the terms: -12 + t = -8.485281374 Solving -12 + t = -8.485281374 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + t = -8.485281374 + 12 Combine like terms: -12 + 12 = 0 0 + t = -8.485281374 + 12 t = -8.485281374 + 12 Combine like terms: -8.485281374 + 12 = 3.514718626 t = 3.514718626 Simplifying t = 3.514718626

Solution

The solution to the problem is based on the solutions from the subproblems. t = {20.485281374, 3.514718626}

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