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solve each equation by completing the square

To solve the quadratic equation \( x^2 + 8x + 3 = 0 \) by completing the square, follow these steps:

### Step 1: Move the constant to the other side
We want to isolate the \(x\)-terms on one side of the equation:
\[
x^2 + 8x = -3
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides. The coefficient of \(x\) is 8, so:
\[
\left(\frac{8}{2}\right)^2 = 16
\]
Now add 16 to both sides of the equation:
\[
x^2 + 8x + 16 = -3 + 16
\]
\[
x^2 + 8x + 16 = 13
\]

### Step 3: Rewrite as a perfect square trinomial
The left side of the equation is now a perfect square trinomial:
\[
(x + 4)^2 = 13
\]

### Step 4: Solve for \(x\)
To solve for \(x\), take the square root of both sides:
\[
x + 4 = \pm \sqrt{13}
\]

### Step 5: Isolate \(x\)
Finally, solve for \(x\) by subtracting 4 from both sides:
\[
x = -4 \pm \sqrt{13}
\]

Thus, the solutions are:
\[
x = -4 + \sqrt{13} \quad \text{or} \quad x = -4 - \sqrt{13}
\]

Would you like more details on any of the steps? Let me know!

Here are 5 related questions to deepen your understanding:
1. How do you complete the square for equations with a coefficient on \(x^2\) other than 1?
2. What is the geometric interpretation of completing the square?
3. How can you solve the same equation using the quadratic formula?
4. How is completing the square useful in deriving the quadratic formula?
5. What are the real-world applications of quadratic equations?

**Tip:** When completing the square, always ensure the coefficient of \(x^2\) is 1 before proceeding with the method.

Solve each equation by completing the square: x^2 + 8x + 3 = 0

Learn how to solve the quadratic equation x^2 + 8x + 3 = 0 step-by-step using the method of completing the square. This tutorial explains key algebraic concepts, including quadratic equations and properties of square roots. Perfect for students in grades 9-12 looking to improve their problem-solving skills.

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