Question

Solve for x.
16x2–16x+3=0
Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.
x=

Answer 1

To solve the quadratic equation \( 16x^2 - 16x + 3 = 0 \), we can use the quadratic formula:

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

In this equation, \( a = 16 \), \( b = -16 \), and \( c = 3 \).

1. Calculate the discriminant (\( b^2 - 4ac \)):
\[
b^2 = (-16)^2 = 256
\]
\[
4ac = 4 \cdot 16 \cdot 3 = 192
\]
\[
b^2 - 4ac = 256 - 192 = 64
\]

2. Now we can substitute into the quadratic formula:
\[
x = \frac{-(-16) \pm \sqrt{64}}{2 \cdot 16} = \frac{16 \pm 8}{32}
\]

3. This gives us two potential solutions:
- For \( 16 + 8 \):
\[
x = \frac{24}{32} = \frac{3}{4}
\]
- For \( 16 - 8 \):
\[
x = \frac{8}{32} = \frac{1}{4}
\]

Thus, the solutions for the equation \( 16x^2 - 16x + 3 = 0 \) are:

\[
x = \frac{3}{4}, \frac{1}{4}
\]

Therefore, the final answer is:

\[
x = \frac{3}{4}, \frac{1}{4}
\]