Answer
The length is $3$
Work Step by Step
Let's consider a trinomial in this form: $~x^2+bx+c$
To factor this trinomial, we need to find two numbers $r$ and $s$ such that $r+s = b$ and $r\times s = c$
Then we can factor the trinomial as follows:
$~x^2+bx+c = (x+r)~(x+s)$
We can rearrange the given equation:
$x~(x+5)=24$
$x^2+5x=24$
$x^2+5x-24 = 0$
To factor the left side of the equation, we need to find two numbers $r$ and $s$ such that $r+s = 5$ and $r\times s = -24$. We can see that $(8)+(-3) = 5~$ and $(8)\times (-3) = -24$
We can solve the equation as follows:
$x~(x+5)=24$
$x^2+5x=24$
$x^2+5x-24 = 0$
$(x+8)~(x-3) = 0$
$x+8 = 0~~$ or $~~x-3 = 0$
$x = -8~~$ or $~~x = 3$
Since the length must be a positive number, the length is $3$
