Question

solve for p.
42p² + 91p - 56 = 0
write each solution as an integer, proper fraction, or improper fraction in simplest form.
if there are multiple solutions, separate them with commas.
p =
submit

Explanation:

Step1: Use quadratic formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 42\), \(b = 91\), \(c=-56\).
First, calculate the discriminant \(\Delta=b^2 - 4ac=(91)^2-4\times42\times(-56)\)
\(= 8281+4\times42\times56\)
\(= 8281 + 9408\)
\(= 17689\)
\(\sqrt{\Delta}=\sqrt{17689} = 133\)

Step2: Find the roots

Now, find the two roots:
\(p_1=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-91 + 133}{2\times42}=\frac{42}{84}=\frac{1}{2}\)
\(p_2=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-91-133}{2\times42}=\frac{-224}{84}=-\frac{8}{3}\)

Answer:

\(\frac{1}{2},-\frac{8}{3}\)