Question

use the quadratic formula to solve. express your answer in simplest form.
$8p^2 + 14p - 9 = 6$

Explanation:

Step1: Rewrite the equation in standard form

Subtract 6 from both sides of the equation \(8p^{2}+14p - 9=6\) to get \(8p^{2}+14p-15 = 0\). For a quadratic equation \(ax^{2}+bx + c=0\), here \(a = 8\), \(b=14\), \(c=- 15\).

Step2: Recall the quadratic formula

The quadratic formula is \(p=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).

Step3: Calculate the discriminant \(\Delta=b^{2}-4ac\)

Substitute \(a = 8\), \(b = 14\), \(c=-15\) into the discriminant formula:
\(\Delta=(14)^{2}-4\times8\times(-15)=196 + 480=676\)

Step4: Substitute into the quadratic formula

\(p=\frac{-14\pm\sqrt{676}}{2\times8}=\frac{-14\pm26}{16}\)
We have two cases:
Case 1: \(p=\frac{-14 + 26}{16}=\frac{12}{16}=\frac{3}{4}\)
Case 2: \(p=\frac{-14-26}{16}=\frac{-40}{16}=-\frac{5}{2}\)

Answer:

\(p=\frac{3}{4}\) or \(p =-\frac{5}{2}\)