t^2=3t+5

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Solution for t^2=3t+5 equation:


Simplifying
t2 = 3t + 5

Reorder the terms:
t2 = 5 + 3t

Solving
t2 = 5 + 3t

Solving for variable 't'.

Reorder the terms:
-5 + -3t + t2 = 5 + 3t + -5 + -3t

Reorder the terms:
-5 + -3t + t2 = 5 + -5 + 3t + -3t

Combine like terms: 5 + -5 = 0
-5 + -3t + t2 = 0 + 3t + -3t
-5 + -3t + t2 = 3t + -3t

Combine like terms: 3t + -3t = 0
-5 + -3t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -3t + 5 + t2 = 0 + 5

Reorder the terms:
-5 + 5 + -3t + t2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -3t + t2 = 0 + 5
-3t + t2 = 0 + 5

Combine like terms: 0 + 5 = 5
-3t + t2 = 5

The t term is -3t.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3t + 2.25 + t2 = 5 + 2.25

Reorder the terms:
2.25 + -3t + t2 = 5 + 2.25

Combine like terms: 5 + 2.25 = 7.25
2.25 + -3t + t2 = 7.25

Factor a perfect square on the left side:
(t + -1.5)(t + -1.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(t + -1.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

t + -1.5 = 2.692582404 Simplifying t + -1.5 = 2.692582404 Reorder the terms: -1.5 + t = 2.692582404 Solving -1.5 + t = 2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 2.692582404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 2.692582404 + 1.5 t = 2.692582404 + 1.5 Combine like terms: 2.692582404 + 1.5 = 4.192582404 t = 4.192582404 Simplifying t = 4.192582404

Subproblem 2

t + -1.5 = -2.692582404 Simplifying t + -1.5 = -2.692582404 Reorder the terms: -1.5 + t = -2.692582404 Solving -1.5 + t = -2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -2.692582404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -2.692582404 + 1.5 t = -2.692582404 + 1.5 Combine like terms: -2.692582404 + 1.5 = -1.192582404 t = -1.192582404 Simplifying t = -1.192582404

Solution

The solution to the problem is based on the solutions from the subproblems. t = {4.192582404, -1.192582404}

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