To factor the quadratic equation \( f(x) = x^2 - 5x - 6 \), we need to find two numbers that multiply to \(-6\) (the constant term) and add up to \(-5\) (the coefficient of the linear term \(x\)).
The two numbers that satisfy these conditions are \(-6\) and \(1\), since:
- \(-6 \cdot 1 = -6\)
- \(-6 + 1 = -5\)
Thus, we can write the factored form of the quadratic as: \[ f(x) = (x - 6)(x + 1) \]
This corresponds to option C: \((x + 1)(x - 6)\).
Verification through Expansion
To verify the factorization, we can expand \((x + 1)(x - 6)\): \[ (x + 1)(x - 6) = x^2 - 6x + 1x - 6 = x^2 - 5x - 6 \] This confirms that the factorization is correct.
Graph Verification
To verify graphically, you can plot the function \(f(x) = x^2 - 5x - 6\) and observe the points where the graph intersects the x-axis (the roots). The roots of the quadratic based on our factorization are \(x = 6\) and \(x = -1\).
In practice, you could use graphing software or a graphing calculator. Here's a brief description of how you could visualize it:
- Create the graph for \(y = x^2 - 5x - 6\).
- Look for the points where the graph intersects the x-axis at \(x = 6\) and \(x = -1\).
- Confirm that these points correspond to the factors found.
Conclusion
The correct factored form is \((x + 1)(x - 6)\), which is option C.