Two sides of a triangle have lengths 3 and 5. The length of the third side is a root of the equation x squared minus 6x plus 8 equals 0. Find the perimeter of the triangle.

Step-by-step solution

1. 1

   Solve the quadratic to get the candidates. Factor by finding two numbers with product $8$ and sum $-6$:

   $x^{2}-6x+8=(x-2)(x-4)=0\ \Longrightarrow\ x=2\ \text{or}\ x=4.$

   Both are positive, so both survive the obvious "a length must be positive" filter - and that is exactly the trap this problem is built around.

2. 2

   Recall the constraint the quadratic knows nothing about. Three positive numbers form a triangle only if each side is shorter than the sum of the other two. With the two known sides $3$ and $5$, the third side $x$ must satisfy

   $5-3<x<5+3,\qquad\text{that is}\qquad 2<x<8.$

3. 3

   Test $x=2$. Here $3+2=5$, which is equal to the remaining side rather than greater than it. The three segments would lie flat in a straight line, giving a degenerate figure with no area, so $x=2$ must be rejected.

4. 4

   Test $x=4$. Check all three inequalities: $3+5=8>4$, $3+4=7>5$, and $5+4=9>3$. All hold strictly, so a genuine triangle with sides $3$, $4$, $5$ exists - in fact the familiar right triangle, since $3^2+4^2=5^2$.

5. 5

   Compute the perimeter. Only one root is admissible, so there is a single answer:

   $P=3+5+4=12.$

   Answering "$10$ or $12$" is the standard mistake: it comes from using both roots without applying the triangle inequality.