5(t-3)-2t^2=-30

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Solution for 5(t-3)-2t^2=-30 equation:


Simplifying
5(t + -3) + -2t2 = -30

Reorder the terms:
5(-3 + t) + -2t2 = -30
(-3 * 5 + t * 5) + -2t2 = -30
(-15 + 5t) + -2t2 = -30

Solving
-15 + 5t + -2t2 = -30

Solving for variable 't'.

Reorder the terms:
-15 + 30 + 5t + -2t2 = -30 + 30

Combine like terms: -15 + 30 = 15
15 + 5t + -2t2 = -30 + 30

Combine like terms: -30 + 30 = 0
15 + 5t + -2t2 = 0

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

Divide each side by '-2'.
-7.5 + -2.5t + t2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + -2.5t + 7.5 + t2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + -2.5t + t2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + -2.5t + t2 = 0 + 7.5
-2.5t + t2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
-2.5t + t2 = 7.5

The t term is -2.5t.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5t + 1.5625 + t2 = 7.5 + 1.5625

Reorder the terms:
1.5625 + -2.5t + t2 = 7.5 + 1.5625

Combine like terms: 7.5 + 1.5625 = 9.0625
1.5625 + -2.5t + t2 = 9.0625

Factor a perfect square on the left side:
(t + -1.25)(t + -1.25) = 9.0625

Calculate the square root of the right side: 3.010398645

Break this problem into two subproblems by setting 
(t + -1.25) equal to 3.010398645 and -3.010398645.

Subproblem 1

t + -1.25 = 3.010398645 Simplifying t + -1.25 = 3.010398645 Reorder the terms: -1.25 + t = 3.010398645 Solving -1.25 + t = 3.010398645 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 3.010398645 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 3.010398645 + 1.25 t = 3.010398645 + 1.25 Combine like terms: 3.010398645 + 1.25 = 4.260398645 t = 4.260398645 Simplifying t = 4.260398645

Subproblem 2

t + -1.25 = -3.010398645 Simplifying t + -1.25 = -3.010398645 Reorder the terms: -1.25 + t = -3.010398645 Solving -1.25 + t = -3.010398645 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -3.010398645 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -3.010398645 + 1.25 t = -3.010398645 + 1.25 Combine like terms: -3.010398645 + 1.25 = -1.760398645 t = -1.760398645 Simplifying t = -1.760398645

Solution

The solution to the problem is based on the solutions from the subproblems. t = {4.260398645, -1.760398645}

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