Etiennne is graphing the quadratic equation `y=x^(2)-8x-48`. He substtitute 0 for x and finds that the y-intercept of the graph is-48. Next, he wants to plot the x-intercepts of the graph, so he rewrites the equation in a different form. Assuming he rewrote the equations correctly and the equation reveals the x-intercepts, which of the following Etienne's new equations?

Options

A. `Y=(x-8)^(2)-16`
B. `y=(x-8)^(2)+16`
C. `y=(x-8)(x+6)`
D. `y=(x-12)(x+4)`

Explanation

To solve the problem, we need to rewrite the quadratic equation \( y = x^2 - 8x - 48 \) in a form that reveals the x-intercepts. Here’s how we can do it step by step:

### Step 1: Set the equation to find x-intercepts
To find the x-intercepts, we set \( y = 0 \):
\[ 0 = x^2 - 8x - 48 \]

### Step 2: Rearrange the equation
This gives us the quadratic equation:
\[ x^2 - 8x - 48 = 0 \]

### Step 3: Factor the quadratic equation
Next, we need to factor the quadratic expression. We are looking for two numbers that multiply to \(-48\) (the constant term) and add to \(-8\) (the coefficient of \(x\)). The numbers that satisfy this are \(-12\) and \(4\).

So we can rewrite the equation:
\[ x^2 - 12x + 4x - 48 = 0 \]

### Step 4: Group the terms
Now, we can group the terms:
\[ (x^2 - 12x) + (4x - 48) = 0 \]

### Step 5: Factor by grouping
Factoring out the common terms from each group:
\[ x(x - 12) + 4(x - 12) = 0 \]

Now we can factor out \((x - 12)\):
\[ (x - 12)(x + 4) = 0 \]

### Step 6: Solve for x-intercepts
Setting each factor to zero gives us the x-intercepts:
\[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \]
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]

Thus, the x-intercepts are \( (12, 0) \) and \( (-4, 0) \).

### Final Answer
Etienne's new equation that reveals the x-intercepts is:
\[ (x - 12)(x + 4) = 0 \]