(3t^2-2t+7)+(-4t^2+5t+1)=0

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Solution for (3t^2-2t+7)+(-4t^2+5t+1)=0 equation:


Simplifying
(3t2 + -2t + 7) + (-4t2 + 5t + 1) = 0

Reorder the terms:
(7 + -2t + 3t2) + (-4t2 + 5t + 1) = 0

Remove parenthesis around (7 + -2t + 3t2)
7 + -2t + 3t2 + (-4t2 + 5t + 1) = 0

Reorder the terms:
7 + -2t + 3t2 + (1 + 5t + -4t2) = 0

Remove parenthesis around (1 + 5t + -4t2)
7 + -2t + 3t2 + 1 + 5t + -4t2 = 0

Reorder the terms:
7 + 1 + -2t + 5t + 3t2 + -4t2 = 0

Combine like terms: 7 + 1 = 8
8 + -2t + 5t + 3t2 + -4t2 = 0

Combine like terms: -2t + 5t = 3t
8 + 3t + 3t2 + -4t2 = 0

Combine like terms: 3t2 + -4t2 = -1t2
8 + 3t + -1t2 = 0

Solving
8 + 3t + -1t2 = 0

Solving for variable 't'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-8 + -3t + t2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -3t + 8 + t2 = 0 + 8

Reorder the terms:
-8 + 8 + -3t + t2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -3t + t2 = 0 + 8
-3t + t2 = 0 + 8

Combine like terms: 0 + 8 = 8
-3t + t2 = 8

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 = 8 + 2.25

Reorder the terms:
2.25 + -3t + t2 = 8 + 2.25

Combine like terms: 8 + 2.25 = 10.25
2.25 + -3t + t2 = 10.25

Factor a perfect square on the left side:
(t + -1.5)(t + -1.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(t + -1.5) equal to 3.201562119 and -3.201562119.

Subproblem 1

t + -1.5 = 3.201562119 Simplifying t + -1.5 = 3.201562119 Reorder the terms: -1.5 + t = 3.201562119 Solving -1.5 + t = 3.201562119 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 = 3.201562119 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 3.201562119 + 1.5 t = 3.201562119 + 1.5 Combine like terms: 3.201562119 + 1.5 = 4.701562119 t = 4.701562119 Simplifying t = 4.701562119

Subproblem 2

t + -1.5 = -3.201562119 Simplifying t + -1.5 = -3.201562119 Reorder the terms: -1.5 + t = -3.201562119 Solving -1.5 + t = -3.201562119 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 = -3.201562119 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -3.201562119 + 1.5 t = -3.201562119 + 1.5 Combine like terms: -3.201562119 + 1.5 = -1.701562119 t = -1.701562119 Simplifying t = -1.701562119

Solution

The solution to the problem is based on the solutions from the subproblems. t = {4.701562119, -1.701562119}

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