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!