Solve the following equations using the quadratic formula: 1) 3x² + 4x − 2 = 0, 2) 2x² − 3x − 7 = 0, 3) 3q² − 5q − 5 = 0. 4a) Show that x−3 = 5/3x reduces to 3x² − 9x − 5 = 0. 4b) Solve the equation 3x² − 9x − 5 = 0

For the quadratic equation $ax^2 + bx + c = 0$, we have $a=3$, $b=4$, and $c=-2$.

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

Divide all terms by 2:

The two solutions are:

The solutions are $\boxed{x = \frac{-2 \pm \sqrt{10}}{3}}$.

2) Solve $2x^2 - 3x - 7 = 0$

For $ax^2 + bx + c = 0$, we have $a=2$, $b=-3$, and $c=-7$.

Substitute the values:

The two solutions are:

The solutions are $\boxed{x = \frac{3 \pm \sqrt{65}}{4}}$.

3) Solve $3q^2 - 5q - 5 = 0$

For $aq^2 + bq + c = 0$, we have $a=3$, $b=-5$, and $c=-5$.

Substitute the values:

The two solutions are:

The solutions are $\boxed{q = \frac{5 \pm \sqrt{85}}{6}}$.

4a) Show that $x-3 = \frac{5}{3x}$ reduces to $3x^2 - 9x - 5 = 0$

Subtract 5 from both sides:

This shows that the equation $x-3 = \frac{5}{3x}$ reduces to $\boxed{3x^2 - 9x - 5 = 0}$.

4b) Solve the equation $3x^2 - 9x - 5 = 0$

For $ax^2 + bx + c = 0$, we have $a=3$, $b=-9$, and $c=-5$.

Substitute the values:

The two solutions are:

The solutions are $\boxed{x = \frac{9 \pm \sqrt{141}}{6}}$.