Find the real solutions of the equation (3x - 5)^2 - 8(3x - 5) + 16 = 0.

Step-by-step solution

1. 1

   Identify the repeated expression. The binomial $3x-5$ occurs squared and to the first power, so a substitution collapses the equation to a plain quadratic.

2. 2

   Set u = 3x - 5. The equation becomes $u^2-8u+16=0$. Note $16=4^2$ and $8=2\cdot 4$, which is exactly the pattern of a perfect square trinomial.

3. 3

   Factor as a perfect square. $u^2-8u+16=(u-4)^2$, so $(u-4)^2=0$ forces $u=4$. Because the factor is repeated, this is a double root rather than two distinct values.

4. 4

   Solve for x. From $3x-5=4$ we get $3x=9$, so $x=3$. There is only one solution, and it has multiplicity two.

5. 5

   Substitute back. With $x=3$: $3x-5=4$, so $4^2-8(4)+16=16-32+16=0$. The left side is exactly zero, confirming the single repeated root.