Question

solve for z.
\\( 3z^2 - 11z - 4 = 0 \\)
write each solution as an integer, proper fraction, or improper fraction in simplest form. if there are multiple solutions, separate them with commas.
\\( z = \\)

Explanation:

Step1: Factor the quadratic equation

We have the quadratic equation \(3z^2 - 11z - 4 = 0\). We need to factor it. We look for two numbers that multiply to \(3\times(-4)= - 12\) and add up to \(-11\). The numbers are \(-12\) and \(1\).
So we can rewrite the middle term:
\(3z^2-12z + z-4 = 0\)
Now factor by grouping:
\(3z(z - 4)+1(z - 4)=0\)
\((3z + 1)(z - 4)=0\)

Step2: Solve for z using zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
Case 1: \(3z+1 = 0\)
Subtract \(1\) from both sides: \(3z=-1\)
Divide both sides by \(3\): \(z=-\frac{1}{3}\)
Case 2: \(z - 4=0\)
Add \(4\) to both sides: \(z = 4\)

Answer:

\(4,-\frac{1}{3}\)