10=24t-5t^2

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Solution for 10=24t-5t^2 equation:


Simplifying
10 = 24t + -5t2

Solving
10 = 24t + -5t2

Solving for variable 't'.

Reorder the terms:
10 + -24t + 5t2 = 24t + -24t + -5t2 + 5t2

Combine like terms: 24t + -24t = 0
10 + -24t + 5t2 = 0 + -5t2 + 5t2
10 + -24t + 5t2 = -5t2 + 5t2

Combine like terms: -5t2 + 5t2 = 0
10 + -24t + 5t2 = 0

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

Divide each side by '5'.
2 + -4.8t + t2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -4.8t + -2 + t2 = 0 + -2

Reorder the terms:
2 + -2 + -4.8t + t2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -4.8t + t2 = 0 + -2
-4.8t + t2 = 0 + -2

Combine like terms: 0 + -2 = -2
-4.8t + t2 = -2

The t term is -4.8t.  Take half its coefficient (-2.4).
Square it (5.76) and add it to both sides.

Add '5.76' to each side of the equation.
-4.8t + 5.76 + t2 = -2 + 5.76

Reorder the terms:
5.76 + -4.8t + t2 = -2 + 5.76

Combine like terms: -2 + 5.76 = 3.76
5.76 + -4.8t + t2 = 3.76

Factor a perfect square on the left side:
(t + -2.4)(t + -2.4) = 3.76

Calculate the square root of the right side: 1.939071943

Break this problem into two subproblems by setting 
(t + -2.4) equal to 1.939071943 and -1.939071943.

Subproblem 1

t + -2.4 = 1.939071943 Simplifying t + -2.4 = 1.939071943 Reorder the terms: -2.4 + t = 1.939071943 Solving -2.4 + t = 1.939071943 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.4' to each side of the equation. -2.4 + 2.4 + t = 1.939071943 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + t = 1.939071943 + 2.4 t = 1.939071943 + 2.4 Combine like terms: 1.939071943 + 2.4 = 4.339071943 t = 4.339071943 Simplifying t = 4.339071943

Subproblem 2

t + -2.4 = -1.939071943 Simplifying t + -2.4 = -1.939071943 Reorder the terms: -2.4 + t = -1.939071943 Solving -2.4 + t = -1.939071943 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.4' to each side of the equation. -2.4 + 2.4 + t = -1.939071943 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + t = -1.939071943 + 2.4 t = -1.939071943 + 2.4 Combine like terms: -1.939071943 + 2.4 = 0.460928057 t = 0.460928057 Simplifying t = 0.460928057

Solution

The solution to the problem is based on the solutions from the subproblems. t = {4.339071943, 0.460928057}

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