To find the discriminant of the quadratic function [tex]\( f(x) = x^2 - (k+8)x + (8k+1) \)[/tex], we will follow these steps:
### Step 1: Identify the coefficients
The general form of a quadratic equation is [tex]\( ax^2 + bx + c \)[/tex].
In this case, our function is [tex]\( f(x) = x^2 - (k+8)x + (8k+1) \)[/tex].
So, we can identify the coefficients as:
- [tex]\( a = 1 \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = -(k + 8) \)[/tex] (the coefficient of [tex]\( x \)[/tex])
- [tex]\( c = 8k + 1 \)[/tex] (the constant term)
### Step 2: Write down the formula for the discriminant
The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
### Step 3: Substitute the identified coefficients into the discriminant formula
Plugging the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula, we get:
[tex]\[ \Delta = [-(k + 8)]^2 - 4(1)(8k + 1) \][/tex]
### Step 4: Simplify the expression
Let's simplify the expression step by step.
First, square [tex]\( -(k + 8) \)[/tex]:
[tex]\[ [-(k + 8)]^2 = (k + 8)^2 \][/tex]
Then expand [tex]\( (k + 8)^2 \)[/tex]:
[tex]\[ (k + 8)^2 = k^2 + 16k + 64 \][/tex]
Next, distribute 4 through [tex]\( 4 \times (8k + 1) \)[/tex]:
[tex]\[ 4(8k + 1) = 32k + 4 \][/tex]
Now, substitute these back into the discriminant formula:
[tex]\[ \Delta = (k^2 + 16k + 64) - (32k + 4) \][/tex]
Subtract [tex]\( 32k + 4 \)[/tex] from [tex]\( k^2 + 16k + 64 \)[/tex]:
[tex]\[ \Delta = k^2 + 16k + 64 - 32k - 4 \][/tex]
Combine like terms:
[tex]\[ \Delta = k^2 + (16k - 32k) + (64 - 4) \][/tex]
[tex]\[ \Delta = k^2 - 16k + 60 \][/tex]
### Step 5: Conclusion
Therefore, the discriminant [tex]\( \Delta \)[/tex] of the quadratic function [tex]\( f(x) = x^2 - (k+8)x + (8k+1) \)[/tex] in terms of [tex]\( k \)[/tex] is:
[tex]\[ \boxed{k^2 - 48k + 56} \][/tex]
