(-3t^2+4t-7)+(6t^2+t+2)=0

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Solution for (-3t^2+4t-7)+(6t^2+t+2)=0 equation:


Simplifying
(-3t2 + 4t + -7) + (6t2 + t + 2) = 0

Reorder the terms:
(-7 + 4t + -3t2) + (6t2 + t + 2) = 0

Remove parenthesis around (-7 + 4t + -3t2)
-7 + 4t + -3t2 + (6t2 + t + 2) = 0

Reorder the terms:
-7 + 4t + -3t2 + (2 + t + 6t2) = 0

Remove parenthesis around (2 + t + 6t2)
-7 + 4t + -3t2 + 2 + t + 6t2 = 0

Reorder the terms:
-7 + 2 + 4t + t + -3t2 + 6t2 = 0

Combine like terms: -7 + 2 = -5
-5 + 4t + t + -3t2 + 6t2 = 0

Combine like terms: 4t + t = 5t
-5 + 5t + -3t2 + 6t2 = 0

Combine like terms: -3t2 + 6t2 = 3t2
-5 + 5t + 3t2 = 0

Solving
-5 + 5t + 3t2 = 0

Solving for variable 't'.

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

Divide each side by '3'.
-1.666666667 + 1.666666667t + t2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667t + 1.666666667 + t2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + 1.666666667t + t2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + 1.666666667t + t2 = 0 + 1.666666667
1.666666667t + t2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
1.666666667t + t2 = 1.666666667

The t term is 1.666666667t.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667t + 0.6944444447 + t2 = 1.666666667 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667t + t2 = 1.666666667 + 0.6944444447

Combine like terms: 1.666666667 + 0.6944444447 = 2.3611111117
0.6944444447 + 1.666666667t + t2 = 2.3611111117

Factor a perfect square on the left side:
(t + 0.8333333335)(t + 0.8333333335) = 2.3611111117

Calculate the square root of the right side: 1.536590743

Break this problem into two subproblems by setting 
(t + 0.8333333335) equal to 1.536590743 and -1.536590743.

Subproblem 1

t + 0.8333333335 = 1.536590743 Simplifying t + 0.8333333335 = 1.536590743 Reorder the terms: 0.8333333335 + t = 1.536590743 Solving 0.8333333335 + t = 1.536590743 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + t = 1.536590743 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + t = 1.536590743 + -0.8333333335 t = 1.536590743 + -0.8333333335 Combine like terms: 1.536590743 + -0.8333333335 = 0.7032574095 t = 0.7032574095 Simplifying t = 0.7032574095

Subproblem 2

t + 0.8333333335 = -1.536590743 Simplifying t + 0.8333333335 = -1.536590743 Reorder the terms: 0.8333333335 + t = -1.536590743 Solving 0.8333333335 + t = -1.536590743 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + t = -1.536590743 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + t = -1.536590743 + -0.8333333335 t = -1.536590743 + -0.8333333335 Combine like terms: -1.536590743 + -0.8333333335 = -2.3699240765 t = -2.3699240765 Simplifying t = -2.3699240765

Solution

The solution to the problem is based on the solutions from the subproblems. t = {0.7032574095, -2.3699240765}

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