Explanation

To find the roots of the quadratic equation x² - 7x - 9 = 0:

Use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

In this case, a = 1, b = -7, and c = -9:

x = (7 ± √((-7)² - 4(1)(-9))) / 2(1)

= (7 ± √(49 + 36)) / 2

= (7 ± √85) / 2

The two roots are:

x₁ = (7 + √85) / 2

x₂ = (7 - √85) / 2

The positive difference between the roots is:

|x₁ - x₂| = |((7 + √85) / 2) - ((7 - √85) / 2)|

= |(7 + √85 - 7 + √85) / 2|

= |2√85 / 2|

= √85