Amira is solving the equation *x*2 – 6*x* = 1. Which value must be added to both sides of the equation to make the left side a perfect-square trinomial?


–9

8

9

36

To solve this problem, let's complete the square. We start with the given equation:

\[
x^2 - 6x = 1
\]

### Step 1: Completing the square
To make the left-hand side a perfect square trinomial, we need to add a value to both sides of the equation. The formula to complete the square is:

\[
\left(\frac{b}{2}\right)^2
\]

Where \(b\) is the coefficient of \(x\) in the quadratic expression.

In this case, \(b = -6\). Let's compute:

\[
\left(\frac{-6}{2}\right)^2 = (-3)^2 = 9
\]

So, we need to **add 9** to both sides of the equation.

### Step 2: Add to both sides
Adding 9 to both sides:

\[
x^2 - 6x + 9 = 1 + 9
\]

Now, the left-hand side is a perfect square trinomial, which can be factored as:

\[
(x - 3)^2 = 10
\]

### Conclusion:
The value that must be added to both sides is **9**.

The correct answer is **9**.

---

Would you like further details or have any questions? Here are some related questions for further practice:

1. How do you solve a quadratic equation by completing the square?
2. What is the general form of a perfect-square trinomial?
3. How can you factor a perfect-square trinomial?
4. Can every quadratic equation be solved by completing the square?
5. What are the benefits of completing the square in solving quadratic equations?

**Tip:** Completing the square is useful not only for solving quadratic equations but also for deriving the quadratic formula!

Amira is solving the equation x^2 – 6x = 1. Which value must be added to both sides of the equation to make the left side a perfect-square trinomial? Options: –9, 8, 9, 36.

Learn how to complete the square for the quadratic equation x^2 - 6x = 1. This tutorial explains the process, including adding 9 to both sides to form a perfect-square trinomial. Perfect for students in Grades 8-10 looking to master quadratic equations.

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