The equation `1/4(4x ^(2) -8x-k) =30` has two solutions: `x =-5 and x =7.` What is the value of 2k ?

Explanation

To solve the equation \( \frac{1}{4}(4x^2 - 8x - k) = 30 \) given that it has solutions \( x = -5 \) and \( x = 7 \), we can follow these steps:

### Step 1: Substitute one of the solutions into the equation
Let's substitute \( x = 7 \) into the equation.

\[ \frac{1}{4}(4(7)^2 - 8(7) - k) = 30 \]

### Step 2: Simplify the equation
Calculating \( 4(7)^2 \) and \( 8(7) \):

\[ 4(7^2) = 4 \times 49 = 196 \]
\[ 8(7) = 56 \]

Now substitute these values back into the equation:

\[ \frac{1}{4}(196 - 56 - k) = 30 \]

### Step 3: Combine like terms
Now simplify the expression inside the parentheses:

\[ 196 - 56 = 140 \]

So the equation becomes:

\[ \frac{1}{4}(140 - k) = 30 \]

### Step 4: Eliminate the fraction
Multiply both sides of the equation by 4 to eliminate the fraction:

\[ 140 - k = 120 \]

### Step 5: Solve for \( k \)
Now, isolate \( k \):

\[ -k = 120 - 140 \]
\[ -k = -20 \]
\[ k = 20 \]

### Step 6: Find \( 2k \)
Now we need to find \( 2k \):

\[ 2k = 2 \times 20 = 40 \]

### Final Answer
The value of \( 2k \) is \( 40 \).

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