To factor the quadratic expression \( f^2 + 15f - 15 \), we can use the method of factoring by grouping, or we can apply the quadratic formula if necessary.
First, we look for two numbers that multiply to \( -15 \) (the constant term) and add up to \( 15 \) (the coefficient of \( f \)). However, since the numbers that meet these conditions don't seem to exist, we will apply the quadratic formula instead:
The quadratic formula is given by:
\[ f = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 15 \), and \( c = -15 \).
- Calculate the discriminant:
\[ b^2 - 4ac = 15^2 - 4(1)(-15) = 225 + 60 = 285 \]
- Now apply the quadratic formula:
\[ f = \frac{-15 \pm \sqrt{285}}{2(1)} = \frac{-15 \pm \sqrt{285}}{2} \]
- Since \( \sqrt{285} \) can be simplified. We can check for perfect square factors of 285:
\[ 285 = 3 \times 95 = 3 \times 5 \times 19 \]
Thus, \( \sqrt{285} \) cannot be simplified further.
- The expression does not factor neatly with integers. The factored form in terms of its roots is:
\[ f^2 + 15f - 15 = (f - \frac{-15 + \sqrt{285}}{2})(f - \frac{-15 - \sqrt{285}}{2}) \]
So, it does not factor nicely into integers or rational numbers. The roots are approximately \( -0.177 \) and \( -14.823 \).
If you need a numerical approximation:
- One root is approximately \( -0.177 \)
- The other root is approximately \( -14.823 \)
So the expression is:
\[ f^2 + 15f - 15 = (f - r_1)(f - r_2) \]
where \( r_1 \) and \( r_2 \) are those roots.