solve the quadratic equation by completing the squares x^2 + 6x -1 = 0 checking

To solve the quadratic equation \(x^2 + 6x - 1 = 0\) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side
Start by isolating the \(x^2\) and \(x\) terms on one side of the equation:
\[
x^2 + 6x = 1
\]

### 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:
\[
x^2 + 6x + 9 = 1 + 9
\]

This simplifies to:
\[
(x + 3)^2 = 10
\]

### Step 3: Solve for \(x\)
To solve for \(x\), take the square root of both sides:
\[
x + 3 = \pm \sqrt{10}
\]

Subtract 3 from both sides to solve for \(x\):
\[
x = -3 \pm \sqrt{10}
\]

### Step 4: Final answer
The solutions to the quadratic equation are:
\[
x = -3 + \sqrt{10} \quad \text{or} \quad x = -3 - \sqrt{10}
\]

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

1. **For \(x = -3 + \sqrt{10}\)**:
   \[
   \left(-3 + \sqrt{10}\right)^2 + 6\left(-3 + \sqrt{10}\right) - 1
   \]
   Simplifying:
   \[
   (9 - 6\sqrt{10} + 10) + (-18 + 6\sqrt{10}) - 1 = 19 - 19 = 0
   \]

2. **For \(x = -3 - \sqrt{10}\)**:
   \[
   \left(-3 - \sqrt{10}\right)^2 + 6\left(-3 - \sqrt{10}\right) - 1
   \]
   Simplifying:
   \[
   (9 + 6\sqrt{10} + 10) + (-18 - 6\sqrt{10}) - 1 = 19 - 19 = 0
   \]

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. What is the quadratic formula, and how does it relate to completing the square?
2. How can you derive the quadratic formula using the method of completing the square?
3. What are the advantages and disadvantages of completing the square versus other methods for solving quadratic equations?
4. How can you apply the method of completing the square to solve quadratic inequalities?
5. How does the discriminant in the quadratic formula relate to the number of real solutions?

**Tip:** Completing the square is not only a method for solving quadratics but also a way to convert the quadratic expression into its vertex form, which can be useful in graphing.

Learn how to solve the quadratic equation x^2 + 6x - 1 = 0 using the method of completing the square. This detailed step-by-step solution covers key algebraic concepts, including completing the square and the quadratic formula. Suitable for students in Grades 10-12.

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