The equation `(36^(2)+16x-21)/(tx -4) =-9x+5 -(1)/(tx -4)` is true for all values of x for which `x ne 4/t,` where t is a constant. What is the value of t ?

Options

A. `-20`
B. `-4`
C. 4
D. 12

Answer

To solve the equation

\[ \frac{36x^2 + 16x - 21}{tx - 4} = -9x + 5 - \frac{1}{tx - 4} \]

we will follow these steps:

### Step 1: Identify the common denominator
The common denominator on the right-hand side is \(tx - 4\). We can rewrite the right-hand side as:

\[ -9x + 5 - \frac{1}{tx - 4} = \frac{(-9x + 5)(tx - 4) - 1}{tx - 4} \]

### Step 2: Multiply both sides by the common denominator
To eliminate the denominators, we multiply both sides by \(tx - 4\):

\[ 36x^2 + 16x - 21 = (-9x + 5)(tx - 4) - 1 \]

### Step 3: Expand the right-hand side
Now, we will expand the right-hand side:

\[ (-9x + 5)(tx - 4) = -9tx^2 + 36x + 5tx - 20 \]

So, the equation becomes:

\[ 36x^2 + 16x - 21 = -9tx^2 + (36 + 5t)x - 20 - 1 \]

### Step 4: Simplify the right-hand side
Combine the constants on the right-hand side:

\[ -20 - 1 = -21 \]

Thus, we have:

\[ 36x^2 + 16x - 21 = -9tx^2 + (36 + 5t)x - 21 \]

### Step 5: Set the numerators equal
Since the denominators are the same, we can equate the numerators:

\[ 36x^2 + 16x - 21 = -9tx^2 + (36 + 5t)x - 21 \]

### Step 6: Cancel out the constant terms
The constant terms \(-21\) on both sides cancel out:

\[ 36x^2 + 16x = -9tx^2 + (36 + 5t)x \]

### Step 7: Compare coefficients
Now we will compare the coefficients of \(x^2\) and \(x\):

1. For \(x^2\) terms:
\[ 36 = -9t \]

2. For \(x\) terms:
\[ 16 = 36 + 5t \]

### Step 8: Solve for \(t\)
From the first equation:

\[ t = \frac{36}{-9} = -4 \]

### Step 9: Verify with the second equation
Substituting \(t = -4\) into the second equation:

\[ 16 = 36 + 5(-4) \]
\[ 16 = 36 - 20 \]
\[ 16 = 16 \]

This confirms that our value for \(t\) is correct.

### Final Answer
Thus, the value of \(t\) is

\[ \boxed{-4} \]