4t^2+8t-42=0

Solution for 4t^2+8t-42=0 equation:

Simplifying
4t2 + 8t + -42 = 0

Reorder the terms:
-42 + 8t + 4t2 = 0

Solving
-42 + 8t + 4t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-21 + 4t + 2t2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-21 + 4t + 2t2)' equal to zero and attempt to solve:

Simplifying
-21 + 4t + 2t2 = 0

Solving
-21 + 4t + 2t2 = 0

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

Divide each side by '2'.
-10.5 + 2t + t2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + 2t + 10.5 + t2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + 2t + t2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + 2t + t2 = 0 + 10.5
2t + t2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
2t + t2 = 10.5

The t term is 2t.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2t + 1 + t2 = 10.5 + 1

Reorder the terms:
1 + 2t + t2 = 10.5 + 1

Combine like terms: 10.5 + 1 = 11.5
1 + 2t + t2 = 11.5

Factor a perfect square on the left side:
(t + 1)(t + 1) = 11.5

Calculate the square root of the right side: 3.391164992

Break this problem into two subproblems by setting 
(t + 1) equal to 3.391164992 and -3.391164992.

Subproblem 1
t + 1 = 3.391164992

Simplifying
t + 1 = 3.391164992

Reorder the terms:
1 + t = 3.391164992

Solving
1 + t = 3.391164992

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + t = 3.391164992 + -1

Combine like terms: 1 + -1 = 0
0 + t = 3.391164992 + -1
t = 3.391164992 + -1

Combine like terms: 3.391164992 + -1 = 2.391164992
t = 2.391164992

Simplifying
t = 2.391164992

Subproblem 2
t + 1 = -3.391164992

Simplifying
t + 1 = -3.391164992

Reorder the terms:
1 + t = -3.391164992

Solving
1 + t = -3.391164992

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + t = -3.391164992 + -1

Combine like terms: 1 + -1 = 0
0 + t = -3.391164992 + -1
t = -3.391164992 + -1

Combine like terms: -3.391164992 + -1 = -4.391164992
t = -4.391164992

Simplifying
t = -4.391164992

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.391164992, -4.391164992}
Solution
t = {2.391164992, -4.391164992}