Find all real solutions of the quadratic equation. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.)
16x2 − 8x + 1 = 0

We can solve the quadratic equation \( 16x^2 - 8x + 1 = 0 \) by using the quadratic formula:

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

For the equation \( 16x^2 - 8x + 1 = 0 \), we have the following values:

- \( a = 16 \)
- \( b = -8 \)
- \( c = 1 \)

Now, we substitute these values into the quadratic formula.

### Step 1: Calculate the discriminant

\[
\text{Discriminant} = b^2 - 4ac = (-8)^2 - 4(16)(1) = 64 - 64 = 0
\]

### Step 2: Apply the quadratic formula

Since the discriminant is zero, there is one real solution (a double root):

\[
x = \frac{-(-8) \pm \sqrt{0}}{2(16)} = \frac{8 \pm 0}{32} = \frac{8}{32} = \frac{1}{4}
\]

Thus, the solution is:

\[
x = \frac{1}{4}
\]

### Final Answer:
There is one real solution, \( x = \frac{1}{4} \).

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Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the solution change if the discriminant were positive?
2. Can a quadratic equation have no real solutions?
3. How does completing the square compare to using the quadratic formula?
4. What is the relationship between the discriminant and the nature of the solutions?
5. How does the graph of \( y = 16x^2 - 8x + 1 \) look, and where is its vertex?

**Tip:** If the discriminant is zero, the quadratic equation has exactly one real solution (a repeated root).

Find all real solutions of the quadratic equation. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.) 16x^2 − 8x + 1 = 0

Explore how to solve the quadratic equation 16x² − 8x + 1 = 0 using the quadratic formula. This detailed guide covers discriminant analysis and explains why there is one real solution, x = 1/4. Suitable for high school students (Grades 9-12).

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