4x^2-8x+3=0

To solve the quadratic equation \( 4x^2 - 8x + 3 = 0 \), we can use the quadratic formula:

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

In this equation, \( a = 4 \), \( b = -8 \), and \( c = 3 \).

1. Calculate the discriminant \( b^2 - 4ac \):

\[ b^2 = (-8)^2 = 64 \] \[ 4ac = 4 \cdot 4 \cdot 3 = 48 \] \[ b^2 - 4ac = 64 - 48 = 16 \]

2. Now, substitute the values back into the quadratic formula:

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

3. Now, calculate the two possible values for \( x \):

\[ x_1 = \frac{8 + 4}{8} = \frac{12}{8} = \frac{3}{2} \] \[ x_2 = \frac{8 - 4}{8} = \frac{4}{8} = \frac{1}{2} \]

Thus, the solutions to the equation \( 4x^2 - 8x + 3 = 0 \) are:

\[ x = \frac{3}{2} \quad \text{and} \quad x = \frac{1}{2} \]