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).
The numbers that satisfy these conditions are \(-6\) and \(1\). This is because:
- \(-6 \times 1 = -6\)
- \(-6 + 1 = -5\)
Thus, we can factor the quadratic as follows:
\[ f(x) = (x - 6)(x + 1) \]
Now let's verify this factorization by expanding it back:
\[ (x - 6)(x + 1) = x^2 + x - 6x - 6 = x^2 - 5x - 6 \]
This matches the original function, confirming the factorization is correct.
Graphical Verification
To further verify this, you can graph the function \( f(x) = x^2 - 5x - 6 \) and observe where the graph intersects the x-axis. The x-intercepts should correspond to the roots of the equation, which are \( x = 6 \) and \( x = -1 \).
Using a graphing tool or software:
- Graph \( f(x) = x^2 - 5x - 6 \).
- Look for the points where the graph intersects the x-axis. These points should be \( (6, 0) \) and \( (-1, 0) \).
Thus, the correct factored form is \( (x - 6)(x + 1) \) and the graphical representation will confirm the roots.