(4t^2-2t-6)+(3t^2-t+5)=0

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Solution for (4t^2-2t-6)+(3t^2-t+5)=0 equation:


Simplifying
(4t2 + -2t + -6) + (3t2 + -1t + 5) = 0

Reorder the terms:
(-6 + -2t + 4t2) + (3t2 + -1t + 5) = 0

Remove parenthesis around (-6 + -2t + 4t2)
-6 + -2t + 4t2 + (3t2 + -1t + 5) = 0

Reorder the terms:
-6 + -2t + 4t2 + (5 + -1t + 3t2) = 0

Remove parenthesis around (5 + -1t + 3t2)
-6 + -2t + 4t2 + 5 + -1t + 3t2 = 0

Reorder the terms:
-6 + 5 + -2t + -1t + 4t2 + 3t2 = 0

Combine like terms: -6 + 5 = -1
-1 + -2t + -1t + 4t2 + 3t2 = 0

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

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

Solving
-1 + -3t + 7t2 = 0

Solving for variable 't'.

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

Divide each side by '7'.
-0.1428571429 + -0.4285714286t + t2 = 0

Move the constant term to the right:

Add '0.1428571429' to each side of the equation.
-0.1428571429 + -0.4285714286t + 0.1428571429 + t2 = 0 + 0.1428571429

Reorder the terms:
-0.1428571429 + 0.1428571429 + -0.4285714286t + t2 = 0 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + -0.4285714286t + t2 = 0 + 0.1428571429
-0.4285714286t + t2 = 0 + 0.1428571429

Combine like terms: 0 + 0.1428571429 = 0.1428571429
-0.4285714286t + t2 = 0.1428571429

The t term is -0.4285714286t.  Take half its coefficient (-0.2142857143).
Square it (0.04591836735) and add it to both sides.

Add '0.04591836735' to each side of the equation.
-0.4285714286t + 0.04591836735 + t2 = 0.1428571429 + 0.04591836735

Reorder the terms:
0.04591836735 + -0.4285714286t + t2 = 0.1428571429 + 0.04591836735

Combine like terms: 0.1428571429 + 0.04591836735 = 0.18877551025
0.04591836735 + -0.4285714286t + t2 = 0.18877551025

Factor a perfect square on the left side:
(t + -0.2142857143)(t + -0.2142857143) = 0.18877551025

Calculate the square root of the right side: 0.434483038

Break this problem into two subproblems by setting 
(t + -0.2142857143) equal to 0.434483038 and -0.434483038.

Subproblem 1

t + -0.2142857143 = 0.434483038 Simplifying t + -0.2142857143 = 0.434483038 Reorder the terms: -0.2142857143 + t = 0.434483038 Solving -0.2142857143 + t = 0.434483038 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.2142857143 + t = 0.434483038 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + t = 0.434483038 + 0.2142857143 t = 0.434483038 + 0.2142857143 Combine like terms: 0.434483038 + 0.2142857143 = 0.6487687523 t = 0.6487687523 Simplifying t = 0.6487687523

Subproblem 2

t + -0.2142857143 = -0.434483038 Simplifying t + -0.2142857143 = -0.434483038 Reorder the terms: -0.2142857143 + t = -0.434483038 Solving -0.2142857143 + t = -0.434483038 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.2142857143 + t = -0.434483038 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + t = -0.434483038 + 0.2142857143 t = -0.434483038 + 0.2142857143 Combine like terms: -0.434483038 + 0.2142857143 = -0.2201973237 t = -0.2201973237 Simplifying t = -0.2201973237

Solution

The solution to the problem is based on the solutions from the subproblems. t = {0.6487687523, -0.2201973237}

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