25(t)+56=40(t)+3(t^2)

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Solution for 25(t)+56=40(t)+3(t^2) equation:


Simplifying
25(t) + 56 = 40(t) + 3(t2)

Reorder the terms:
56 + 25t = 40(t) + 3(t2)

Solving
56 + 25t = 40t + 3t2

Solving for variable 't'.

Combine like terms: 25t + -40t = -15t
56 + -15t + -3t2 = 40t + 3t2 + -40t + -3t2

Reorder the terms:
56 + -15t + -3t2 = 40t + -40t + 3t2 + -3t2

Combine like terms: 40t + -40t = 0
56 + -15t + -3t2 = 0 + 3t2 + -3t2
56 + -15t + -3t2 = 3t2 + -3t2

Combine like terms: 3t2 + -3t2 = 0
56 + -15t + -3t2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-18.66666667 + 5t + t2 = 0

Move the constant term to the right:

Add '18.66666667' to each side of the equation.
-18.66666667 + 5t + 18.66666667 + t2 = 0 + 18.66666667

Reorder the terms:
-18.66666667 + 18.66666667 + 5t + t2 = 0 + 18.66666667

Combine like terms: -18.66666667 + 18.66666667 = 0.00000000
0.00000000 + 5t + t2 = 0 + 18.66666667
5t + t2 = 0 + 18.66666667

Combine like terms: 0 + 18.66666667 = 18.66666667
5t + t2 = 18.66666667

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 = 18.66666667 + 6.25

Reorder the terms:
6.25 + 5t + t2 = 18.66666667 + 6.25

Combine like terms: 18.66666667 + 6.25 = 24.91666667
6.25 + 5t + t2 = 24.91666667

Factor a perfect square on the left side:
(t + 2.5)(t + 2.5) = 24.91666667

Calculate the square root of the right side: 4.991659711

Break this problem into two subproblems by setting 
(t + 2.5) equal to 4.991659711 and -4.991659711.

Subproblem 1

t + 2.5 = 4.991659711 Simplifying t + 2.5 = 4.991659711 Reorder the terms: 2.5 + t = 4.991659711 Solving 2.5 + t = 4.991659711 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 = 4.991659711 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 4.991659711 + -2.5 t = 4.991659711 + -2.5 Combine like terms: 4.991659711 + -2.5 = 2.491659711 t = 2.491659711 Simplifying t = 2.491659711

Subproblem 2

t + 2.5 = -4.991659711 Simplifying t + 2.5 = -4.991659711 Reorder the terms: 2.5 + t = -4.991659711 Solving 2.5 + t = -4.991659711 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 = -4.991659711 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -4.991659711 + -2.5 t = -4.991659711 + -2.5 Combine like terms: -4.991659711 + -2.5 = -7.491659711 t = -7.491659711 Simplifying t = -7.491659711

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.491659711, -7.491659711}

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