For [x^2 - 9x - 22 = 0] solve the quadratic equation by completing the square. 

To solve the quadratic equation \( x^2 - 9x - 22 = 0 \) by completing the square, follow these steps:

### Step 1: Move the constant to the other side
First, move the constant term \(-22\) to the right-hand side of the equation:
\[
x^2 - 9x = 22
\]

### Step 2: Complete the square
To complete the square, we need to add a number to both sides that makes the left-hand side a perfect square trinomial. The coefficient of \(x\) is \(-9\). Take half of this coefficient, square it, and add it to both sides:

\[
\left(\frac{-9}{2}\right)^2 = \left(\frac{-9}{2}\right)^2 = \frac{81}{4}
\]

Now, add \(\frac{81}{4}\) to both sides of the equation:

\[
x^2 - 9x + \frac{81}{4} = 22 + \frac{81}{4}
\]

### Step 3: Simplify the right-hand side
On the right-hand side, we need to combine \(22\) and \(\frac{81}{4}\). Convert 22 into a fraction with a denominator of 4:

\[
22 = \frac{88}{4}
\]

Now, add \(\frac{88}{4}\) and \(\frac{81}{4}\):

\[
\frac{88}{4} + \frac{81}{4} = \frac{169}{4}
\]

So the equation becomes:

\[
x^2 - 9x + \frac{81}{4} = \frac{169}{4}
\]

### Step 4: Factor the left-hand side
The left-hand side is now a perfect square trinomial, and it factors as:

\[
\left(x - \frac{9}{2}\right)^2 = \frac{169}{4}
\]

### Step 5: Solve for \(x\)
Take the square root of both sides:

\[
x - \frac{9}{2} = \pm \frac{\sqrt{169}}{2} = \pm \frac{13}{2}
\]

### Step 6: Isolate \(x\)
Now, solve for \(x\) by adding \(\frac{9}{2}\) to both sides:

\[
x = \frac{9}{2} \pm \frac{13}{2}
\]

This gives two solutions:

1. \( x = \frac{9}{2} + \frac{13}{2} = \frac{22}{2} = 11 \)
2. \( x = \frac{9}{2} - \frac{13}{2} = \frac{-4}{2} = -2 \)

### Final Answer:
The solutions are:
\[
x = 11 \quad \text{or} \quad x = -2
\]

Would you like more details on any of the steps? Here are some related questions:

1. What is the general method for completing the square for any quadratic equation?
2. How does completing the square relate to the quadratic formula?
3. Can you solve quadratic equations graphically by completing the square?
4. What are some practical applications of completing the square?
5. How do the solutions of this method compare to factoring or using the quadratic formula?

**Tip:** Completing the square is also useful when converting a quadratic equation to vertex form.

For [x^2 - 9x - 22 = 0] solve the quadratic equation by completing the square.

Learn how to solve the quadratic equation x^2 - 9x - 22 = 0 by completing the square. This step-by-step solution covers essential algebra concepts like completing the square and properties of perfect square trinomials, suitable for Grades 9-11 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation