Given F(x)=x^2 - x - 4. If f(x)=8, what is the value of x?

You are given the function \( F(x) = x^2 - x - 4 \) and need to find the value of \( x \) when \( f(x) = 8 \).

To solve for \( x \), set \( F(x) \) equal to 8:

\[
x^2 - x - 4 = 8
\]

Subtract 8 from both sides to set the equation to zero:

\[
x^2 - x - 12 = 0
\]

Now, solve the quadratic equation \( x^2 - x - 12 = 0 \).

To factor it, find two numbers that multiply to \(-12\) and add to \(-1\). The numbers are 3 and \(-4\).

Thus, the factorization is:

\[
(x - 4)(x + 3) = 0
\]

Now, solve for \( x \):

1. \( x - 4 = 0 \) gives \( x = 4 \)
2. \( x + 3 = 0 \) gives \( x = -3 \)

Therefore, the solutions for \( x \) are \( x = 4 \) and \( x = -3 \).

Would you like further details on any step or have any questions?

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**Relative questions:**
1. What if \( f(x) = -8 \) instead of 8?
2. How can the quadratic formula be applied to solve this equation?
3. What happens to the solution if the constant term changes to a different value?
4. How would the graph of \( F(x) = x^2 - x - 4 \) look?
5. How would you verify these solutions using substitution?

**Tip:** Factoring quadratics works well when the factors are easy to identify, but always remember the quadratic formula can be a reliable backup.

Learn how to solve the quadratic equation x^2 - x - 4 = 8 step-by-step. This problem involves finding roots using factoring and the quadratic formula, covering key algebraic concepts. Suitable for high school students.

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