Solve the following quations by using qardratic formula:
`256x^(2)-32x+1=0`

Answer

To solve the quadratic equation \(256x^2 - 32x + 1 = 0\) using the quadratic formula, we will follow these steps:

### Step 1: Identify coefficients
We start by identifying the coefficients \(a\), \(b\), and \(c\) from the standard form of a quadratic equation \(ax^2 + bx + c = 0\).

- Here, \(a = 256\)
- \(b = -32\)
- \(c = 1\)

### Step 2: Write the quadratic formula
The quadratic formula is given by:

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

### Step 3: Substitute the values into the formula
We will now substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula.

\[ x = \frac{-(-32) \pm \sqrt{(-32)^2 - 4 \cdot 256 \cdot 1}}{2 \cdot 256} \]

### Step 4: Simplify the expression
First, simplify \(-(-32)\):

\[ x = \frac{32 \pm \sqrt{1024 - 1024}}{512} \]

Now calculate \(b^2 - 4ac\):

\[ (-32)^2 = 1024 \]
\[ 4 \cdot 256 \cdot 1 = 1024 \]
Thus, \(b^2 - 4ac = 1024 - 1024 = 0\).

### Step 5: Substitute back into the formula
Now substitute back into the formula:

\[ x = \frac{32 \pm \sqrt{0}}{512} \]

### Step 6: Calculate the roots
Since \(\sqrt{0} = 0\), we have:

\[ x = \frac{32}{512} \]

### Step 7: Simplify the fraction
Now simplify \(\frac{32}{512}\):

\[ x = \frac{1}{16} \]

### Conclusion
Thus, the solution to the quadratic equation \(256x^2 - 32x + 1 = 0\) is:

\[ x = \frac{1}{16} \]

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