roots of the quadratic equation 9x2+6x+1=0

To find the roots of the quadratic equation \(9x^2 + 6x + 1 = 0\), we can use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the equation \(9x^2 + 6x + 1 = 0\), the coefficients are:
- \(a = 9\)
- \(b = 6\)
- \(c = 1\)

### Step-by-Step Solution:

1. **Calculate the discriminant (\( \Delta \)):**

\[ \Delta = b^2 - 4ac \]

Substitute the values of \(a\), \(b\), and \(c\):

\[ \Delta = 6^2 - 4 \cdot 9 \cdot 1 \]
\[ \Delta = 36 - 36 \]
\[ \Delta = 0 \]

2. **Since the discriminant is zero (\( \Delta = 0 \)), there is exactly one real root (a repeated root).**

3. **Calculate the root using the quadratic formula:**

\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]

Since \(\Delta = 0\), this simplifies to:

\[ x = \frac{-b}{2a} \]

Substitute \(b = 6\) and \(a = 9\):

\[ x = \frac{-6}{2 \cdot 9} \]
\[ x = \frac{-6}{18} \]
\[ x = \frac{-1}{3} \]

### Conclusion:

The quadratic equation \(9x^2 + 6x + 1 = 0\) has a single repeated root:

\[ x = -\frac{1}{3} \]

Would you like more details or have any questions?

Here are 5 related questions you might ask:
1. How do you find the vertex of a quadratic function?
2. What is the significance of the discriminant in a quadratic equation?
3. Can you show an example of solving a quadratic equation by completing the square?
4. How do you graph a quadratic function?
5. What are the applications of quadratic equations in real life?

**Tip:** The discriminant (\( \Delta = b^2 - 4ac \)) of a quadratic equation determines the nature of the roots. If \(\Delta > 0\), there are two distinct real roots. If \(\Delta = 0\), there is one repeated real root. If \(\Delta < 0\), there are two complex conjugate roots.

Discover how to find the roots of the quadratic equation 9x^2 + 6x + 1 = 0 using the quadratic formula. This detailed solution explains the concept of discriminant and its impact on the nature of roots. Suitable for high school students.

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