t^2=5t+3

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


Simplifying
t2 = 5t + 3

Reorder the terms:
t2 = 3 + 5t

Solving
t2 = 3 + 5t

Solving for variable 't'.

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

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

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The t term is -5t.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5t + 6.25 + t2 = 3 + 6.25

Reorder the terms:
6.25 + -5t + t2 = 3 + 6.25

Combine like terms: 3 + 6.25 = 9.25
6.25 + -5t + t2 = 9.25

Factor a perfect square on the left side:
(t + -2.5)(t + -2.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(t + -2.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

t + -2.5 = 3.041381265 Simplifying t + -2.5 = 3.041381265 Reorder the terms: -2.5 + t = 3.041381265 Solving -2.5 + t = 3.041381265 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = 3.041381265 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 3.041381265 + 2.5 t = 3.041381265 + 2.5 Combine like terms: 3.041381265 + 2.5 = 5.541381265 t = 5.541381265 Simplifying t = 5.541381265

Subproblem 2

t + -2.5 = -3.041381265 Simplifying t + -2.5 = -3.041381265 Reorder the terms: -2.5 + t = -3.041381265 Solving -2.5 + t = -3.041381265 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = -3.041381265 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -3.041381265 + 2.5 t = -3.041381265 + 2.5 Combine like terms: -3.041381265 + 2.5 = -0.541381265 t = -0.541381265 Simplifying t = -0.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.541381265, -0.541381265}

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