If the roots of the equation x² - bx + c = 5 differ by 5, then which one of the following is correct ?  

Concept:

1. For the quadratic equation ax² + bx + c = 0

Sum of root (α + β) = -b/a

Product of root = c/a

2. (a - b)² = (a + b)² - 4ab

Calculation:

Given that 

x2 - bx + c = 5

⇒ x² - bx + c - 5 = 0                 

Let the roots of the equation be denoted by α and β.

Sum of roots (α + β) = b

Product of roots (αβ) = c - 5

Also given that the roots differ by 5. Therefore:

α - β = 5 

By using the above identity

⇒ (α - β)² = (α + β)² -4αβ 

⇒ 5² = (b)² - 4(c - 5) = 25 

⇒ b² - 4c + 20 = 25

∴ b²  = 4c + 5