9p²+48p-64

To work with the quadratic expression \(9p^2 + 48p - 64\), you may want to factor it, find its roots, or simplify it in some way. Here, I'll show you how to factor it and find the roots.

Step 1: Factor the Quadratic Expression

The expression is in the standard quadratic form \(ax^2 + bx + c\), where:

• \(a = 9\)
• \(b = 48\)
• \(c = -64\)

To factor this expression, we can look for two numbers that multiply to \(a \times c = 9 \times (-64) = -576\) and add to \(b = 48\).

Step 2: Finding the Roots

We can also find the roots of the quadratic using the quadratic formula: \[ p = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]

Inserting the values:

• \(b^2 - 4ac = 48^2 - 4 \times 9 \times (-64)\)
• \(b^2 = 2304\)
• \(4ac = -2304\)

Now we calculate: \[ b^2 - 4ac = 2304 + 2304 = 4608 \]

Now substitute into the quadratic formula: \[ p = \frac{{-48 \pm \sqrt{4608}}}{{18}} \]

Step 3: Simplifying the Expression

Calculating the square root of 4608: \[ \sqrt{4608} \approx 67.87 \]

Now substitute back into the formula: \[ p = \frac{{-48 \pm 67.87}}{{18}} \]

This gives us two possible values for \(p\):

1. \[ p_1 = \frac{{-48 + 67.87}}{{18}} \approx 1.09 \] 2. \[ p_2 = \frac{{-48 - 67.87}}{{18}} \approx -6.45 \]

So, the roots of the quadratic \(9p^2 + 48p - 64\) are approximately \(p \approx 1.09\) and \(p \approx -6.45\).

Conclusion

Thus, the quadratic expression \(9p^2 + 48p - 64\) factors and has roots that can be calculated as shown. If you need any more specific operation or detail about the quadratic expression, feel free to ask!