1+9t-5t^2=3

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Solution for 1+9t-5t^2=3 equation:


Simplifying
1 + 9t + -5t2 = 3

Solving
1 + 9t + -5t2 = 3

Solving for variable 't'.

Reorder the terms:
1 + -3 + 9t + -5t2 = 3 + -3

Combine like terms: 1 + -3 = -2
-2 + 9t + -5t2 = 3 + -3

Combine like terms: 3 + -3 = 0
-2 + 9t + -5t2 = 0

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
0.4 + -1.8t + t2 = 0

Move the constant term to the right:

Add '-0.4' to each side of the equation.
0.4 + -1.8t + -0.4 + t2 = 0 + -0.4

Reorder the terms:
0.4 + -0.4 + -1.8t + t2 = 0 + -0.4

Combine like terms: 0.4 + -0.4 = 0.0
0.0 + -1.8t + t2 = 0 + -0.4
-1.8t + t2 = 0 + -0.4

Combine like terms: 0 + -0.4 = -0.4
-1.8t + t2 = -0.4

The t term is -1.8t.  Take half its coefficient (-0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
-1.8t + 0.81 + t2 = -0.4 + 0.81

Reorder the terms:
0.81 + -1.8t + t2 = -0.4 + 0.81

Combine like terms: -0.4 + 0.81 = 0.41
0.81 + -1.8t + t2 = 0.41

Factor a perfect square on the left side:
(t + -0.9)(t + -0.9) = 0.41

Calculate the square root of the right side: 0.640312424

Break this problem into two subproblems by setting 
(t + -0.9) equal to 0.640312424 and -0.640312424.

Subproblem 1

t + -0.9 = 0.640312424 Simplifying t + -0.9 = 0.640312424 Reorder the terms: -0.9 + t = 0.640312424 Solving -0.9 + t = 0.640312424 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + t = 0.640312424 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + t = 0.640312424 + 0.9 t = 0.640312424 + 0.9 Combine like terms: 0.640312424 + 0.9 = 1.540312424 t = 1.540312424 Simplifying t = 1.540312424

Subproblem 2

t + -0.9 = -0.640312424 Simplifying t + -0.9 = -0.640312424 Reorder the terms: -0.9 + t = -0.640312424 Solving -0.9 + t = -0.640312424 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + t = -0.640312424 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + t = -0.640312424 + 0.9 t = -0.640312424 + 0.9 Combine like terms: -0.640312424 + 0.9 = 0.259687576 t = 0.259687576 Simplifying t = 0.259687576

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.540312424, 0.259687576}

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