9t^2-216t+108=0

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Solution for 9t^2-216t+108=0 equation:


Simplifying
9t2 + -216t + 108 = 0

Reorder the terms:
108 + -216t + 9t2 = 0

Solving
108 + -216t + 9t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '9'.
9(12 + -24t + t2) = 0

Ignore the factor 9.

Subproblem 1

Set the factor '(12 + -24t + t2)' equal to zero and attempt to solve: Simplifying 12 + -24t + t2 = 0 Solving 12 + -24t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-12' to each side of the equation. 12 + -24t + -12 + t2 = 0 + -12 Reorder the terms: 12 + -12 + -24t + t2 = 0 + -12 Combine like terms: 12 + -12 = 0 0 + -24t + t2 = 0 + -12 -24t + t2 = 0 + -12 Combine like terms: 0 + -12 = -12 -24t + t2 = -12 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 = -12 + 144 Reorder the terms: 144 + -24t + t2 = -12 + 144 Combine like terms: -12 + 144 = 132 144 + -24t + t2 = 132 Factor a perfect square on the left side: (t + -12)(t + -12) = 132 Calculate the square root of the right side: 11.489125293 Break this problem into two subproblems by setting (t + -12) equal to 11.489125293 and -11.489125293.

Subproblem 1

t + -12 = 11.489125293 Simplifying t + -12 = 11.489125293 Reorder the terms: -12 + t = 11.489125293 Solving -12 + t = 11.489125293 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 = 11.489125293 + 12 Combine like terms: -12 + 12 = 0 0 + t = 11.489125293 + 12 t = 11.489125293 + 12 Combine like terms: 11.489125293 + 12 = 23.489125293 t = 23.489125293 Simplifying t = 23.489125293

Subproblem 2

t + -12 = -11.489125293 Simplifying t + -12 = -11.489125293 Reorder the terms: -12 + t = -11.489125293 Solving -12 + t = -11.489125293 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 = -11.489125293 + 12 Combine like terms: -12 + 12 = 0 0 + t = -11.489125293 + 12 t = -11.489125293 + 12 Combine like terms: -11.489125293 + 12 = 0.510874707 t = 0.510874707 Simplifying t = 0.510874707

Solution

The solution to the problem is based on the solutions from the subproblems. t = {23.489125293, 0.510874707}

Solution

t = {23.489125293, 0.510874707}

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