t(t+5)=2t+7

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Solution for t(t+5)=2t+7 equation:


Simplifying
t(t + 5) = 2t + 7

Reorder the terms:
t(5 + t) = 2t + 7
(5 * t + t * t) = 2t + 7
(5t + t2) = 2t + 7

Reorder the terms:
5t + t2 = 7 + 2t

Solving
5t + t2 = 7 + 2t

Solving for variable 't'.

Reorder the terms:
-7 + 5t + -2t + t2 = 7 + 2t + -7 + -2t

Combine like terms: 5t + -2t = 3t
-7 + 3t + t2 = 7 + 2t + -7 + -2t

Reorder the terms:
-7 + 3t + t2 = 7 + -7 + 2t + -2t

Combine like terms: 7 + -7 = 0
-7 + 3t + t2 = 0 + 2t + -2t
-7 + 3t + t2 = 2t + -2t

Combine like terms: 2t + -2t = 0
-7 + 3t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 3t + 7 + t2 = 0 + 7

Reorder the terms:
-7 + 7 + 3t + t2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 3t + t2 = 0 + 7
3t + t2 = 0 + 7

Combine like terms: 0 + 7 = 7
3t + t2 = 7

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 = 7 + 2.25

Reorder the terms:
2.25 + 3t + t2 = 7 + 2.25

Combine like terms: 7 + 2.25 = 9.25
2.25 + 3t + t2 = 9.25

Factor a perfect square on the left side:
(t + 1.5)(t + 1.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(t + 1.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

t + 1.5 = 3.041381265 Simplifying t + 1.5 = 3.041381265 Reorder the terms: 1.5 + t = 3.041381265 Solving 1.5 + t = 3.041381265 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 = 3.041381265 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 3.041381265 + -1.5 t = 3.041381265 + -1.5 Combine like terms: 3.041381265 + -1.5 = 1.541381265 t = 1.541381265 Simplifying t = 1.541381265

Subproblem 2

t + 1.5 = -3.041381265 Simplifying t + 1.5 = -3.041381265 Reorder the terms: 1.5 + t = -3.041381265 Solving 1.5 + t = -3.041381265 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 = -3.041381265 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -3.041381265 + -1.5 t = -3.041381265 + -1.5 Combine like terms: -3.041381265 + -1.5 = -4.541381265 t = -4.541381265 Simplifying t = -4.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.541381265, -4.541381265}

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