find the roots of the following quadratic equations by completing the square.

1.) x² - 2x = 3

To find the roots of the quadratic equation \( x^2 - 2x = 3 \) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side.
We start by isolating the quadratic and linear terms on one side of the equation:
\[
x^2 - 2x = 3
\]

### Step 2: Add the square of half the coefficient of \(x\) to both sides.
The coefficient of \(x\) is \(-2\). Half of \(-2\) is \(-1\), and the square of \(-1\) is \(1\). Add \(1\) to both sides of the equation:
\[
x^2 - 2x + 1 = 3 + 1
\]
This simplifies to:
\[
(x - 1)^2 = 4
\]

### Step 3: Take the square root of both sides.
To solve for \(x\), take the square root of both sides:
\[
x - 1 = \pm 2
\]

### Step 4: Solve for \(x\).
Now, solve for \(x\) by adding \(1\) to both sides:
\[
x = 1 \pm 2
\]
So, the two possible values of \(x\) are:
\[
x = 1 + 2 = 3
\]
\[
x = 1 - 2 = -1
\]

### Final Answer:
The roots of the quadratic equation \(x^2 - 2x = 3\) are:
\[
x = 3 \quad \text{and} \quad x = -1
\]

Would you like more details on completing the square?

Here are five related questions:
1. How do you complete the square when the coefficient of \(x^2\) is not 1?
2. What is the quadratic formula, and how does it compare to completing the square?
3. Can completing the square always be used to solve any quadratic equation?
4. How can completing the square help in deriving the quadratic formula?
5. What are the geometric interpretations of completing the square?

**Tip**: Always ensure the coefficient of \(x^2\) is 1 before completing the square. If it's not, divide the entire equation by the coefficient first.

find the roots of the following quadratic equations by completing the square. 1.) x² - 2x = 3

Discover how to solve the quadratic equation x² - 2x = 3 using the method of completing the square. This detailed, step-by-step solution covers essential algebraic concepts, including quadratic equations and completing the square. Ideal for students in Grades 8-10.

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