3t^2-5t=7

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Solution for 3t^2-5t=7 equation:


Simplifying
3t2 + -5t = 7

Reorder the terms:
-5t + 3t2 = 7

Solving
-5t + 3t2 = 7

Solving for variable 't'.

Reorder the terms:
-7 + -5t + 3t2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + -5t + 3t2 = 0

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

Divide each side by '3'.
-2.333333333 + -1.666666667t + t2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -1.666666667t + 2.333333333 + t2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -1.666666667t + t2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -1.666666667t + t2 = 0 + 2.333333333
-1.666666667t + t2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-1.666666667t + t2 = 2.333333333

The t term is -1.666666667t.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667t + 0.6944444447 + t2 = 2.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667t + t2 = 2.333333333 + 0.6944444447

Combine like terms: 2.333333333 + 0.6944444447 = 3.0277777777
0.6944444447 + -1.666666667t + t2 = 3.0277777777

Factor a perfect square on the left side:
(t + -0.8333333335)(t + -0.8333333335) = 3.0277777777

Calculate the square root of the right side: 1.740051085

Break this problem into two subproblems by setting 
(t + -0.8333333335) equal to 1.740051085 and -1.740051085.

Subproblem 1

t + -0.8333333335 = 1.740051085 Simplifying t + -0.8333333335 = 1.740051085 Reorder the terms: -0.8333333335 + t = 1.740051085 Solving -0.8333333335 + t = 1.740051085 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + t = 1.740051085 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + t = 1.740051085 + 0.8333333335 t = 1.740051085 + 0.8333333335 Combine like terms: 1.740051085 + 0.8333333335 = 2.5733844185 t = 2.5733844185 Simplifying t = 2.5733844185

Subproblem 2

t + -0.8333333335 = -1.740051085 Simplifying t + -0.8333333335 = -1.740051085 Reorder the terms: -0.8333333335 + t = -1.740051085 Solving -0.8333333335 + t = -1.740051085 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + t = -1.740051085 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + t = -1.740051085 + 0.8333333335 t = -1.740051085 + 0.8333333335 Combine like terms: -1.740051085 + 0.8333333335 = -0.9067177515 t = -0.9067177515 Simplifying t = -0.9067177515

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.5733844185, -0.9067177515}

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