https://file.aiquickdraw.com/mathbot/file/1725613613744f6dfsf5n.jpg

Solving quadratic equation by completing the square 

You have provided a list of quadratic equations. To solve quadratic equations by completing the square, we follow these general steps:

1. Move the constant term to the other side of the equation.
2. If the coefficient of the quadratic term (e.g., \(x^2\)) is not 1, divide through by that coefficient.
3. Take half of the coefficient of the linear term (e.g., \(x\)) and square it.
4. Add this square to both sides of the equation.
5. Factor the left-hand side into a binomial square.
6. Solve for the variable by taking the square root of both sides.

I'll demonstrate how to solve the first equation using this method.

**Equation 1**:  
\[ x^2 + 6x = 16 \]

### Step-by-Step Solution:

1. Move the constant term to the other side:
   \[
   x^2 + 6x = 16
   \]
   
2. Take half of the coefficient of the \(x\)-term (which is 6), divide it by 2, and square it:
   \[
   \left(\frac{6}{2}\right)^2 = 9
   \]
   
3. Add 9 to both sides:
   \[
   x^2 + 6x + 9 = 16 + 9
   \]
   \[
   x^2 + 6x + 9 = 25
   \]

4. Factor the left-hand side:
   \[
   (x + 3)^2 = 25
   \]

5. Solve for \(x\) by taking the square root of both sides:
   \[
   x + 3 = \pm 5
   \]

6. Solve for \(x\):
   \[
   x = -3 \pm 5
   \]
   Thus, \(x = 2\) or \(x = -8\).

### Final Solutions:
- \(x = 2\)
- \(x = -8\)

---

Do you want details on how to solve the other equations similarly?

Here are five related questions for further practice:

1. Can you solve equation 2 using the completing the square method?
2. How would you solve a quadratic equation if the coefficient of \(x^2\) is not 1?
3. What is the geometric interpretation of completing the square?
4. How does completing the square help in deriving the quadratic formula?
5. What changes if the quadratic equation has complex solutions?

**Tip**: Always ensure the coefficient of the \(x^2\)-term is 1 before completing the square for easier calculations.

Solving quadratic equation by completing the square

Explore how to solve quadratic equations by completing the square. This step-by-step guide walks through key algebraic concepts such as factoring, squaring, and solving quadratic equations using algebraic methods. Suitable for high school students, especially in grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation