Question

question 6 · 1 point solve 2z(3z - 5)-1 = 0 by using the quadratic formula. (do not round, enter an exact answer, and use a comma to separate multiple solutions.) provide your answer below:

Explanation:

Step1: Expand the equation

First, expand $2z(3z - 5)-1 = 0$. Using the distributive property $a(b - c)=ab - ac$, we get $6z^{2}-10z - 1=0$.

Step2: Identify coefficients

For a quadratic equation $az^{2}+bz + c = 0$, here $a = 6$, $b=-10$, $c=-1$.

Step3: Apply quadratic formula

The quadratic formula is $z=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Substitute the values of $a$, $b$, and $c$ into the formula:
\[

$$\begin{align*} z&=\frac{-(-10)\pm\sqrt{(-10)^{2}-4\times6\times(-1)}}{2\times6}\\ &=\frac{10\pm\sqrt{100 + 24}}{12}\\ &=\frac{10\pm\sqrt{124}}{12}\\ &=\frac{10\pm2\sqrt{31}}{12}\\ &=\frac{5\pm\sqrt{31}}{6} \end{align*}$$

\]

Answer:

$\frac{5+\sqrt{31}}{6},\frac{5 - \sqrt{31}}{6}$