Answer:
The length of the sign is [tex](4x-5)[/tex] inches
Step-by-step explanation:
We are given that,
Area of the square design = [tex]16x^2-40x+25[/tex]
We will first find the roots of the equation [tex]16x^2-40x+2=0[/tex]
The roots of the quadratic equation [tex]ax^2+bx+c=0[/tex] are given by [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
On comparing, a= 16, b= -40 and c= 25
So, the roots of the equation are,
[tex]x=\frac{40\pm\sqrt{(-40)^2-4\times 16\times 25}}{2\times 16}[/tex]
i.e. [tex]x=\frac{40\pm\sqrt{1600-1600}}{32}[/tex]
i.e. [tex]x=\frac{40\pm\sqrt{0}}{32}[/tex]
i.e. [tex]x=\frac{40}{32}[/tex]
i.e. [tex]x=\frac{5}{4}[/tex]
That is, the factors of the polynomial [tex]16x^2-40x+25[/tex] are [tex](4x-5)[/tex] and [tex]4x-5[/tex].
So, Area of the square design = [tex]16x^2-40x+25[/tex] = [tex](4x-5)^2[/tex]
Since, Area of a square = [tex]Length^2[/tex]
Thus, the length of the sign is [tex](4x-5)[/tex] inches
