$$ -4 y^{2}-2 y+1=0 $$ Use the discriminant, $$ b^{2}-4 a c $$, to determine the number of solutions of the given quadratic equation. Then solve the quadratic ing the formula $$ y=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} $$

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1. The given quadratic equation is

$$
-4 y^2 - 2 y + 1 = 0.
$$

2. Identify the coefficients:

$$
a = -4,\quad b = -2,\quad c = 1.
$$

3. Compute the discriminant using

$$
D = b^2 - 4ac:
$$

$$
D = (-2)^2 - 4(-4)(1) = 4 + 16 = 20.
$$

Since $$ D > 0 $$, the equation has two distinct real solutions.

4. Use the quadratic formula

$$
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

substituting the coefficients:

$$
y = \frac{-(-2) \pm \sqrt{20}}{2(-4)} = \frac{2 \pm \sqrt{20}}{-8}.
$$

5. Simplify $$\sqrt{20}$$:

$$
\sqrt{20} = 2\sqrt{5},
$$

thus

$$
y = \frac{2 \pm 2\sqrt{5}}{-8} = -\frac{2(1 \pm \sqrt{5})}{8} = -\frac{1 \pm \sqrt{5}}{4}.
$$

6. Therefore, the two solutions of the quadratic equation are

$$
y = -\frac{1 + \sqrt{5}}{4} \quad \text{and} \quad y = -\frac{1 - \sqrt{5}}{4}.
$$
The quadratic equation $$-4y^2 - 2y + 1 = 0$$ has two distinct real solutions: $$ y = -\frac{1 + \sqrt{5}}{4} \quad \text{and} \quad y = -\frac{1 - \sqrt{5}}{4}. $$

Use the discriminant, , to determine the number of solutions of the given quadratic equation. Then solve the quadratic
ing the formula

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Use the discriminant, , to determine the number of solutions of the given quadratic equation. Then solve the quadratic ing the formula