Questions: Use the quadratic formula to solve. Express your answer in simplest form. 16 p^2 - 24 p + 5 = 0

Solution Steps

To solve the quadratic equation \(16p^2 - 24p + 5 = 0\) using the quadratic formula, we first identify the coefficients \(a\), \(b\), and \(c\) from the equation \(ax^2 + bx + c = 0\). Then, we apply the quadratic formula:

\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This will give us the solutions for \(p\).

The given quadratic equation is

\[ 16p^2 - 24p + 5 = 0 \]

From this equation, we identify the coefficients as follows:

• \(a = 16\)
• \(b = -24\)
• \(c = 5\)

We calculate the discriminant \(D\) using the formula

\[ D = b^2 - 4ac \]

Substituting the values of \(a\), \(b\), and \(c\):

\[ D = (-24)^2 - 4 \cdot 16 \cdot 5 = 576 - 320 = 256 \]

Using the quadratic formula

\[ p = \frac{-b \pm \sqrt{D}}{2a} \]

we substitute \(D = 256\), \(a = 16\), and \(b = -24\):

\[ p = \frac{-(-24) \pm \sqrt{256}}{2 \cdot 16} = \frac{24 \pm 16}{32} \]

This gives us two potential solutions:

\[ p_1 = \frac{24 + 16}{32} = \frac{40}{32} = 1.25 \]

\[ p_2 = \frac{24 - 16}{32} = \frac{8}{32} = 0.25 \]

Final Answer