F4 mathssssssssssss?

For x² + 5x - k + 4 = 0 has 2 distinct real roots: △ = 5² - 4(- k + 4) > 0
25 - 16 + 4k > 0
k > - 9/4
Let α , β be two real roots and α - β = 13,
(α - β)² = 13²
α² + β² - 2αβ = 169
(α + β)² - 4αβ = 169
(- 5)² - 4(- k + 4) = 169
25 + 4k - 16 = 169
k = 40

Method 2:
Given α - β = 13 and Sum of the roots α + β = - 5 , solved : α = 4 , β = - 9.
The product of roots αβ = 4(- 9) = - 36 = - k + 4 , then k = 40.

x^2+5x-k+4=0 has 2 distinct real roots
i.e. Δ>0
(5)^2-4(1)(-k+4)>0
25+4k-16>0
k>-9/4

the difference between two real roots is 13
i.e. α-β=13
at the same time, α+β=-5 ; αβ=-k+4 (I assume you know why)
α-β=sqrt((α-β)^2)
α-β=sqrt(α^2-2αβ+β^2)
α-β=sqrt(α^2+2αβ+β^2-4αβ)
α-β=sqrt((α+β)^2-4αβ)
13=sqrt((-5)^2-4(-k+4))
169=25+4k-16
160=4k
k=40

Hope I can help you