23t^2-54t-84=0

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Solution for 23t^2-54t-84=0 equation:


Simplifying
23t2 + -54t + -84 = 0

Reorder the terms:
-84 + -54t + 23t2 = 0

Solving
-84 + -54t + 23t2 = 0

Solving for variable 't'.

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

Divide each side by '23'.
-3.652173913 + -2.347826087t + t2 = 0

Move the constant term to the right:

Add '3.652173913' to each side of the equation.
-3.652173913 + -2.347826087t + 3.652173913 + t2 = 0 + 3.652173913

Reorder the terms:
-3.652173913 + 3.652173913 + -2.347826087t + t2 = 0 + 3.652173913

Combine like terms: -3.652173913 + 3.652173913 = 0.000000000
0.000000000 + -2.347826087t + t2 = 0 + 3.652173913
-2.347826087t + t2 = 0 + 3.652173913

Combine like terms: 0 + 3.652173913 = 3.652173913
-2.347826087t + t2 = 3.652173913

The t term is -2.347826087t.  Take half its coefficient (-1.173913044).
Square it (1.378071835) and add it to both sides.

Add '1.378071835' to each side of the equation.
-2.347826087t + 1.378071835 + t2 = 3.652173913 + 1.378071835

Reorder the terms:
1.378071835 + -2.347826087t + t2 = 3.652173913 + 1.378071835

Combine like terms: 3.652173913 + 1.378071835 = 5.030245748
1.378071835 + -2.347826087t + t2 = 5.030245748

Factor a perfect square on the left side:
(t + -1.173913044)(t + -1.173913044) = 5.030245748

Calculate the square root of the right side: 2.242820935

Break this problem into two subproblems by setting 
(t + -1.173913044) equal to 2.242820935 and -2.242820935.

Subproblem 1

t + -1.173913044 = 2.242820935 Simplifying t + -1.173913044 = 2.242820935 Reorder the terms: -1.173913044 + t = 2.242820935 Solving -1.173913044 + t = 2.242820935 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.173913044' to each side of the equation. -1.173913044 + 1.173913044 + t = 2.242820935 + 1.173913044 Combine like terms: -1.173913044 + 1.173913044 = 0.000000000 0.000000000 + t = 2.242820935 + 1.173913044 t = 2.242820935 + 1.173913044 Combine like terms: 2.242820935 + 1.173913044 = 3.416733979 t = 3.416733979 Simplifying t = 3.416733979

Subproblem 2

t + -1.173913044 = -2.242820935 Simplifying t + -1.173913044 = -2.242820935 Reorder the terms: -1.173913044 + t = -2.242820935 Solving -1.173913044 + t = -2.242820935 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.173913044' to each side of the equation. -1.173913044 + 1.173913044 + t = -2.242820935 + 1.173913044 Combine like terms: -1.173913044 + 1.173913044 = 0.000000000 0.000000000 + t = -2.242820935 + 1.173913044 t = -2.242820935 + 1.173913044 Combine like terms: -2.242820935 + 1.173913044 = -1.068907891 t = -1.068907891 Simplifying t = -1.068907891

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.416733979, -1.068907891}

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