120=20t+3t^2

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Solution for 120=20t+3t^2 equation:


Simplifying
120 = 20t + 3t2

Solving
120 = 20t + 3t2

Solving for variable 't'.

Reorder the terms:
120 + -20t + -3t2 = 20t + -20t + 3t2 + -3t2

Combine like terms: 20t + -20t = 0
120 + -20t + -3t2 = 0 + 3t2 + -3t2
120 + -20t + -3t2 = 3t2 + -3t2

Combine like terms: 3t2 + -3t2 = 0
120 + -20t + -3t2 = 0

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

Divide each side by '-3'.
-40 + 6.666666667t + t2 = 0

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + 6.666666667t + 40 + t2 = 0 + 40

Reorder the terms:
-40 + 40 + 6.666666667t + t2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + 6.666666667t + t2 = 0 + 40
6.666666667t + t2 = 0 + 40

Combine like terms: 0 + 40 = 40
6.666666667t + t2 = 40

The t term is 6.666666667t.  Take half its coefficient (3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
6.666666667t + 11.11111112 + t2 = 40 + 11.11111112

Reorder the terms:
11.11111112 + 6.666666667t + t2 = 40 + 11.11111112

Combine like terms: 40 + 11.11111112 = 51.11111112
11.11111112 + 6.666666667t + t2 = 51.11111112

Factor a perfect square on the left side:
(t + 3.333333334)(t + 3.333333334) = 51.11111112

Calculate the square root of the right side: 7.14920353

Break this problem into two subproblems by setting 
(t + 3.333333334) equal to 7.14920353 and -7.14920353.

Subproblem 1

t + 3.333333334 = 7.14920353 Simplifying t + 3.333333334 = 7.14920353 Reorder the terms: 3.333333334 + t = 7.14920353 Solving 3.333333334 + t = 7.14920353 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + t = 7.14920353 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + t = 7.14920353 + -3.333333334 t = 7.14920353 + -3.333333334 Combine like terms: 7.14920353 + -3.333333334 = 3.815870196 t = 3.815870196 Simplifying t = 3.815870196

Subproblem 2

t + 3.333333334 = -7.14920353 Simplifying t + 3.333333334 = -7.14920353 Reorder the terms: 3.333333334 + t = -7.14920353 Solving 3.333333334 + t = -7.14920353 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + t = -7.14920353 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + t = -7.14920353 + -3.333333334 t = -7.14920353 + -3.333333334 Combine like terms: -7.14920353 + -3.333333334 = -10.482536864 t = -10.482536864 Simplifying t = -10.482536864

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.815870196, -10.482536864}

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