If the vertex of the parabola `y=x^(2)-8x+c` lies on x-axis, then the value of c, is

Options

A. -16
B. -4
C. 4
D. 16

Answer

To find the value of \( c \) such that the vertex of the parabola \( y = x^2 - 8x + c \) lies on the x-axis, we can follow these steps:

### Step 1: Identify the vertex of the parabola
The standard form of a parabola is given by \( y = ax^2 + bx + c \). For the given parabola \( y = x^2 - 8x + c \), we can identify \( a = 1 \) and \( b = -8 \).

### Step 2: Calculate the x-coordinate of the vertex
The x-coordinate of the vertex of a parabola can be calculated using the formula:
\[ x = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]

### Step 3: Substitute the x-coordinate back into the equation to find the y-coordinate
Now, we need to find the y-coordinate of the vertex by substituting \( x = 4 \) back into the equation of the parabola:
\[ y = (4)^2 - 8(4) + c \]
Calculating this:
\[ y = 16 - 32 + c = c - 16 \]

### Step 4: Set the y-coordinate equal to 0
Since the vertex lies on the x-axis, the y-coordinate must be 0:
\[ c - 16 = 0 \]

### Step 5: Solve for \( c \)
Now, we can solve for \( c \):
\[ c = 16 \]

Thus, the value of \( c \) is \( \boxed{16} \).

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