Divide the entire equation by the coefficient of x².

The given equation is $2x^2 - 13x + 15 = 0$.
Divide by 2:

$$x^2 - \frac{13}{2}x + \frac{15}{2} = 0$$

The coefficient of $x$ is $-\frac{13}{2}$.
Half of this is $\frac{1}{2} \times \left(-\frac{13}{2}\right) = -\frac{13}{4}$.
Square this value: $\left(-\frac{13}{4}\right)^2 = \frac{169}{16}$.
Add $\frac{169}{16}$ to both sides:

$$x^2 - \frac{13}{2}x + \frac{169}{16} = -\frac{15}{2} + \frac{169}{16}$$

The left side factors as $\left(x - \frac{13}{4}\right)^2$.
For the right side, find a common denominator:

$$-\frac{15}{2} + \frac{169}{16} = -\frac{15 \times 8}{2 \times 8} + \frac{169}{16} = -\frac{120}{16} + \frac{169}{16} = \frac{169 - 120}{16} = \frac{49}{16}$$

So the equation becomes:

$$\left(x - \frac{13}{4}\right)^2 = \frac{49}{16}$$

$$x - \frac{13}{4} = \pm\sqrt{\frac{49}{16}}$$ $$x - \frac{13}{4} = \pm\frac{7}{4}$$

$$x = \frac{13}{4} \pm \frac{7}{4}$$

This gives two possible solutions:

$$x_1 = \frac{13}{4} + \frac{7}{4} = \frac{20}{4} = 5$$ $$x_2 = \frac{13}{4} - \frac{7}{4} = \frac{6}{4} = \frac{3}{2}$$

The solutions are $x=5 \text{ or } x=\frac{3}{2}$.