Understanding the Quadratic Equation
The equation given is a quadratic equation of the form:
\[ 256x^2 - 16x + 16 = 0 \]
In order to solve it, we can use the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
where \( a = 256 \), \( b = -16 \), and \( c = 16 \).
Calculating the Discriminant
First, calculate the discriminant (\( D \)):
- \( D = b^2 - 4ac \)
- \( D = (-16)^2 - 4 \times 256 \times 16 \)
- \( D = 256 - 16384 \)
- \( D = -16128 \)
Analyzing the Discriminant
Since the discriminant \( D \) is negative:
- The equation has no real roots.
- It has two complex roots.
Finding the Complex Roots
To find the roots, we proceed with the quadratic formula:
- Substitute \( D \) back into the formula:
\[ x = \frac{{-(-16) \pm \sqrt{{-16128}}}}{2 \times 256} \]
- Simplifying gives:
\[ x = \frac{{16 \pm i\sqrt{16128}}}{512} \]
- Note that \( \sqrt{16128} \) can be simplified further if required.
Final Roots
Thus, the roots can be expressed as:
- \( x = \frac{16}{512} \pm \frac{i\sqrt{16128}}{512} \)
- Simplifying \( \frac{16}{512} = \frac{1}{32} \)
The final complex roots are:
- \( x = \frac{1}{32} \pm \frac{i\sqrt{16128}}{512} \)
This concludes the solution for the quadratic equation.