-16t^3+100t=72
4t^3 - 25t + 18 = 0
(t-2)(4t^2+8t-9) = 0
To solve the equation -16t^3 + 100t = 72, follow these steps:
Step 1: Move all terms to one side to have the equation equal to zero:
-16t^3 + 100t - 72 = 0
Step 2: Factor out the greatest common factor, if possible:
In this case, there's no common factor to be factored out.
Step 3: Try factoring by grouping or using other factoring techniques:
Unfortunately, the equation does not seem to be easily factorable, so we need to use another method.
Step 4: Use the Rational Root Theorem to find a possible rational root:
The Rational Root Theorem states that any rational root of the equation is of the form p/q, where p is a factor of the constant term (-72) and q is a factor of the leading coefficient (-16).
The possible factors of -72 are ± 1, ± 2, ± 3, ± 4, ± 6, ± 8, ± 9, ± 12, ± 18, ± 24, ± 36, ± 72,
and the possible factors of -16 are ± 1, ± 2, ± 4, ± 8, ± 16.
So, the possible rational roots are:
± 1/1, ± 2/1, ± 3/1, ± 4/1, ± 6/1, ± 8/1, ± 9/1, ± 12/1, ± 18/1, ± 24/1, ± 36/1, ± 72/1,
± 1/2, ± 2/2, ± 3/2, ± 4/2, ± 6/2, ± 8/2, ± 9/2, ± 12/2, ± 18/2, ± 24/2, ± 36/2, ± 72/2,
± 1/4, ± 2/4, ± 3/4, ± 4/4, ± 6/4, ± 8/4, ± 9/4, ± 12/4, ± 18/4, ± 24/4, ± 36/4, ± 72/4,
± 1/8, ± 2/8, ± 3/8, ± 4/8, ± 6/8, ± 8/8, ± 9/8, ± 12/8, ± 18/8, ± 24/8, ± 36/8, ± 72/8,
± 1/16, ± 2/16, ± 3/16, ± 4/16, ± 6/16, ± 8/16, ± 9/16, ± 12/16, ± 18/16, ± 24/16, ± 36/16, ± 72/16.
Step 5: Check each possible rational root using synthetic division or long division:
By testing each possible rational root from the list, we find that t = 3/2 is a root.
Using synthetic division:
```
3/2 | -16 0 100 -72
| -24 -18 63
| -16 -24 82 -9
```
The remainder is -9, indicating that t = 3/2 is not a root.
Step 6: Use numerical methods:
Since the equation is not easily factorable and there are no rational roots, we can use numerical methods like graphing or using a calculator to approximate the solutions.
To solve the equation -16t^3 + 100t = 72, we need to find the values of t that satisfy the equation. Let's go through the steps to find the solution:
Step 1: Rearrange the equation to bring all terms to one side and set it equal to zero:
-16t^3 + 100t - 72 = 0
Step 2: Check if any factors of the constant term (-72) are also factors of the coefficient of the highest power term (-16). In this case, we have to consider both positive and negative factors of 72 (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72) and check if they divide evenly into 16.
Step 3: Test each possible factor in the equation by substituting it for t and checking if the equation is satisfied. If the equation is satisfied, that factor is a solution.
Let's start with the positive factors:
For t = 1:
-16(1)^3 + 100(1) - 72 = -16 + 100 - 72 = 12
Since 12 is not equal to 0, t = 1 is not a solution.
For t = 2:
-16(2)^3 + 100(2) - 72 = -128 + 200 - 72 = 0
Since 0 is equal to 0, t = 2 is a solution.
For t = 3:
-16(3)^3 + 100(3) - 72 = -432 + 300 - 72 = -204
Since -204 is not equal to 0, t = 3 is not a solution.
The process continues for the remaining factors.
For the negative factors: (using the same evaluation as above)
t = -1 is not a solution.
t = -2 is not a solution.
t = -3 is not a solution.
Therefore, the only solution to the equation -16t^3 + 100t = 72 is t = 2.