To solve the quadratic equation:

$2x^2 - 45 = 0$

we need to find the values of $x$ that satisfy this equation.

Step 1: Isolate $x^2$

Divide the entire equation by 2 to simplify:

$x^2 - \frac{45}{2} = 0$

$x^2 = \frac{45}{2}$

Step 2: Take the square root of both sides

To solve for $x$, take the square root of both sides:

$x = \pm \sqrt{\frac{45}{2}}$

Step 3: Simplify the square root

We can simplify $\sqrt{\frac{45}{2}}$ by breaking it down:

[ \sqrt{\frac{45}{2}} = \sqrt{\frac{9 \cdot 5}{2}} = \frac{3\sqrt{10}}{\sqrt{2}} = \frac{3\sqrt{10} \cdot \sqrt{2}}{2} = \frac{3\sqrt{5}}{2}