The equation you've provided is \( y = x^2 - 8x + 16 \).

This is a quadratic equation, and we can analyze it in several ways:

### 1. Factoring:
We can try to factor the quadratic expression:
\[
y = (x - 4)(x - 4) = (x - 4)^2
\]

This shows that the equation represents a perfect square trinomial.

### 2. Vertex Form:
Since the equation is a perfect square, it can be written in vertex form as:
\[
y = (x - 4)^2
\]
The vertex of the parabola is \( (4, 0) \), meaning the graph has its minimum point at \( x = 4 \), and the parabola opens upwards.

### 3. Roots:
To find the roots, we set \( y = 0 \):
\[
(x - 4)^2 = 0
\]
This gives us a single root: \( x = 4 \). The equation has one real root, indicating that the graph touches the x-axis at this point.

### 4. Graph:
The graph of this equation is a parabola that opens upwards, with its vertex at \( (4, 0) \) and no other intersections with the x-axis.

Would you like more details or further explanations?

Here are 5 related questions to expand on this:

1. What is the axis of symmetry of the parabola?
2. How would you complete the square for a general quadratic equation?
3. Can you solve this quadratic using the quadratic formula, and what does it tell you about the discriminant?
4. How does the value of the coefficient of \( x^2 \) affect the shape of a parabola?
5. How do you find the y-intercept of a quadratic equation?

**Tip:** The discriminant of a quadratic equation helps to determine the number and type of roots without solving the equation directly.

Explore how to solve the quadratic equation y = x^2 - 8x + 16 by factoring it as a perfect square trinomial and rewriting it in vertex form. Learn about key concepts such as quadratic equations, the vertex of a parabola, and how to find the roots. Ideal for students in Grades 8-10.

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