A quadratic function f(x) intersects the X-axis at points (6,0) and (8,0). If `f(a)=f(2)=24`,what is the value of `a[ane2]` ?

Answer

To solve the problem step by step, we start with the information given about the quadratic function \( f(x) \).

### Step 1: Write the quadratic function
Since the quadratic function intersects the x-axis at points (6,0) and (8,0), we can express the function in factored form:
\[ f(x) = (x - 6)(x - 8) \]

### Step 2: Expand the function
Now, we expand the factored form to get the standard form of the quadratic function:
\[ f(x) = x^2 - 8x - 6x + 48 = x^2 - 14x + 48 \]

### Step 3: Set up the equation for \( f(2) \)
We know that \( f(2) = 24 \). We can substitute \( x = 2 \) into our function:
\[ f(2) = 2^2 - 14(2) + 48 \]
Calculating this gives:
\[ f(2) = 4 - 28 + 48 = 24 \]
This confirms that our function is correct.

### Step 4: Set up the equation for \( f(a) \)
We also know that \( f(a) = 24 \). Thus, we set up the equation:
\[ f(a) = a^2 - 14a + 48 = 24 \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ a^2 - 14a + 48 - 24 = 0 \]
This simplifies to:
\[ a^2 - 14a + 24 = 0 \]

### Step 6: Factor the quadratic equation
Next, we need to factor the quadratic equation \( a^2 - 14a + 24 = 0 \). We look for two numbers that multiply to 24 and add to -14. The factors are -12 and -2:
\[ (a - 12)(a - 2) = 0 \]

### Step 7: Solve for \( a \)
Setting each factor to zero gives us the possible values for \( a \):
\[ a - 12 = 0 \quad \Rightarrow \quad a = 12 \]
\[ a - 2 = 0 \quad \Rightarrow \quad a = 2 \]

### Step 8: Determine the value of \( a \)
Since we are looking for the value of \( a \) such that \( f(a) = 24 \) and we already know \( f(2) = 24 \), the other value is:
\[ a = 12 \]

Thus, the final answer is:
\[ \boxed{12} \]