3t^2+7t=12

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Solution for 3t^2+7t=12 equation:


Simplifying
3t2 + 7t = 12

Reorder the terms:
7t + 3t2 = 12

Solving
7t + 3t2 = 12

Solving for variable 't'.

Reorder the terms:
-12 + 7t + 3t2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 7t + 3t2 = 0

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

Divide each side by '3'.
-4 + 2.333333333t + t2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 2.333333333t + 4 + t2 = 0 + 4

Reorder the terms:
-4 + 4 + 2.333333333t + t2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 2.333333333t + t2 = 0 + 4
2.333333333t + t2 = 0 + 4

Combine like terms: 0 + 4 = 4
2.333333333t + t2 = 4

The t term is 2.333333333t.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333t + 1.361111112 + t2 = 4 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333t + t2 = 4 + 1.361111112

Combine like terms: 4 + 1.361111112 = 5.361111112
1.361111112 + 2.333333333t + t2 = 5.361111112

Factor a perfect square on the left side:
(t + 1.166666667)(t + 1.166666667) = 5.361111112

Calculate the square root of the right side: 2.315407332

Break this problem into two subproblems by setting 
(t + 1.166666667) equal to 2.315407332 and -2.315407332.

Subproblem 1

t + 1.166666667 = 2.315407332 Simplifying t + 1.166666667 = 2.315407332 Reorder the terms: 1.166666667 + t = 2.315407332 Solving 1.166666667 + t = 2.315407332 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + t = 2.315407332 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + t = 2.315407332 + -1.166666667 t = 2.315407332 + -1.166666667 Combine like terms: 2.315407332 + -1.166666667 = 1.148740665 t = 1.148740665 Simplifying t = 1.148740665

Subproblem 2

t + 1.166666667 = -2.315407332 Simplifying t + 1.166666667 = -2.315407332 Reorder the terms: 1.166666667 + t = -2.315407332 Solving 1.166666667 + t = -2.315407332 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + t = -2.315407332 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + t = -2.315407332 + -1.166666667 t = -2.315407332 + -1.166666667 Combine like terms: -2.315407332 + -1.166666667 = -3.482073999 t = -3.482073999 Simplifying t = -3.482073999

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.148740665, -3.482073999}

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