Certainly! Let's solve the quadratic equation \( 256x^2 - 16x + 16 = 0 \) by splitting the middle term.
Step 1: Identify coefficients
- The quadratic equation is in the form \( ax^2 + bx + c = 0 \).
- Here, \( a = 256 \), \( b = -16 \), and \( c = 16 \).
Step 2: Multiply \( a \) and \( c \)
- Calculate \( a \times c = 256 \times 16 = 4096 \).
Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
- We need two numbers that multiply to \( 4096 \) and add to \( -16 \).
- The numbers are \( -64 \) and \( 48 \) since:
- \( -64 \times 64 = 4096 \)
- \( -64 + 48 = -16 \)
Step 4: Rewrite the equation
- Rewrite the middle term using the found numbers:
\[
256x^2 - 64x + 48x + 16 = 0
\]
Step 5: Group the terms
- Group the first two and the last two terms:
\[
(256x^2 - 64x) + (48x + 16) = 0
\]
Step 6: Factor out common terms
- Factor each group:
\[
64x(4x - 1) + 16(3x + 1) = 0
\]
Step 7: Factor the entire equation
- Combine the factors:
\[
(64x + 16)(4x - 1) = 0
\]
Step 8: Solve for \( x \)
- Set each factor to zero:
1. \( 64x + 16 = 0 \) leads to \( x = -\frac{1}{4} \)
2. \( 4x - 1 = 0 \) leads to \( x = \frac{1}{4} \)
Conclusion
- The solutions are \( x = -\frac{1}{4} \) and \( x = \frac{1}{4} \).