x^{2}-8x+13, find the equation of the axis of symmetry and turning point by completing the square

To find the equation of the axis of symmetry and the turning point by completing the square for the quadratic expression \( x^2 - 8x + 13 \), we follow these steps:

### Step 1: Start with the given expression:
\[
x^2 - 8x + 13
\]

### Step 2: Complete the square.
To complete the square, take the coefficient of \( x \), divide it by 2, and square it. The coefficient of \( x \) is -8, so:

\[
\left( \frac{-8}{2} \right)^2 = (-4)^2 = 16
\]

Now, add and subtract 16 inside the expression:

\[
x^2 - 8x + 16 - 16 + 13 = (x - 4)^2 - 3
\]

### Step 3: Rewrite the expression.
Now, the quadratic expression becomes:

\[
(x - 4)^2 - 3
\]

This is the completed square form.

### Step 4: Find the axis of symmetry and turning point.

- The axis of symmetry occurs at \( x = 4 \), which is the value of \( x \) that makes \( (x - 4) = 0 \).
- The turning point (vertex) is \( (4, -3) \), which comes from the expression \( (x - 4)^2 - 3 \).

### Final Answer:
- The equation of the axis of symmetry is \( x = 4 \).
- The turning point (vertex) is \( (4, -3) \).

Would you like more details on any of these steps or further clarification?

Here are 5 related questions to expand your understanding:
1. How do you find the axis of symmetry for any quadratic function in general form?
2. What is the significance of the vertex in the graph of a quadratic function?
3. How would the turning point change if the quadratic was \( x^2 + 8x + 13 \) instead?
4. Can completing the square be used to solve quadratic equations?
5. How does the discriminant of a quadratic equation relate to the number of real solutions?

**Tip:** Completing the square is a helpful method not only for finding the vertex but also for solving quadratic equations by setting the expression equal to zero.

x^2 - 8x + 13, find the equation of the axis of symmetry and turning point by completing the square.

Learn how to find the axis of symmetry and turning point of the quadratic equation x^2 - 8x + 13 by completing the square. This detailed solution explains algebraic techniques for Grades 8-10 students, covering concepts like completing the square and vertex form.

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