solve the quadratic equation by completing the squares x^2 + 6x + 4= 0 checking

To solve the quadratic equation \(x^2 + 6x + 4 = 0\) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side
First, isolate the \(x^2\) and \(x\) terms on one side of the equation:
\[
x^2 + 6x = -4
\]

### Step 2: Complete the square
To complete the square, take the coefficient of \(x\), divide it by 2, and then square it. The coefficient of \(x\) is 6.

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

Add 9 to both sides of the equation to maintain equality:
\[
x^2 + 6x + 9 = -4 + 9
\]

This simplifies to:
\[
(x + 3)^2 = 5
\]

### Step 3: Solve for \(x\)
To solve for \(x\), take the square root of both sides:
\[
x + 3 = \pm \sqrt{5}
\]

Subtract 3 from both sides to solve for \(x\):
\[
x = -3 \pm \sqrt{5}
\]

### Step 4: Final answer
The solutions to the quadratic equation are:
\[
x = -3 + \sqrt{5} \quad \text{or} \quad x = -3 - \sqrt{5}
\]

### Verification by Substitution
We can verify the solution by substituting these values back into the original equation:

1. **For \(x = -3 + \sqrt{5}\)**:
   \[
   \left(-3 + \sqrt{5}\right)^2 + 6\left(-3 + \sqrt{5}\right) + 4
   \]
   Simplifying:
   \[
   (9 - 6\sqrt{5} + 5) + (-18 + 6\sqrt{5}) + 4 = 14 - 14 = 0
   \]

2. **For \(x = -3 - \sqrt{5}\)**:
   \[
   \left(-3 - \sqrt{5}\right)^2 + 6\left(-3 - \sqrt{5}\right) + 4
   \]
   Simplifying:
   \[
   (9 + 6\sqrt{5} + 5) + (-18 - 6\sqrt{5}) + 4 = 14 - 14 = 0
   \]

Both solutions satisfy the original equation, confirming that the solutions are correct.

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Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How do you graph the solutions of a quadratic equation after solving by completing the square?
2. What is the connection between the vertex form of a quadratic equation and completing the square?
3. How can completing the square help in solving real-world problems modeled by quadratic equations?
4. What is the significance of the discriminant when solving quadratic equations by completing the square?
5. How does completing the square relate to solving quadratic equations with complex solutions?

**Tip:** Completing the square can also help in deriving the quadratic formula, as it effectively transforms any quadratic equation into a form that is easy to solve.

Learn how to solve the quadratic equation x^2 + 6x + 4 = 0 step-by-step using the method of completing the square. This problem involves key algebraic concepts such as completing the square and the quadratic formula. Suitable for high school students in Grades 10-12.

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