You have the quadratic equation:

$4x^2 + 8x + 1 = 0$

To solve this, we can use the quadratic formula:

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

For the equation $4x^2 + 8x + 1 = 0$, the coefficients are:

• $a = 4$
• $b = 8$
• $c = 1$

Let's plug these values into the quadratic formula:

$x = \frac{-8 \pm \sqrt{8^2 - 4(4)(1)}}{2(4)}$

First, calculate the discriminant:

$\sqrt{8^2 - 4(4)(1)} = \sqrt{64 - 16} = \sqrt{48}$

Now simplify $\sqrt{48}$:

$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

So, the equation becomes:

$x = \frac{-8 \pm 4\sqrt{3}}{8}$

Now simplify the expression:

$x = \frac{-8}{8} \pm \frac{4\sqrt{3}}{8}$

$x = -1 \pm \frac{\sqrt{3}}{2}$

Thus, the two solutions are:

$x = -1 + \frac{\sqrt{3}}{2} \quad \text{or} \quad x = -1 - \frac{\sqrt{3}}{2}$

These are the two solutions to the equation $4x^2 + 8x + 1 = 0$.

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Would you like a more detailed explanation on any step? Feel free to ask!

Here are some related questions:

1. How do you complete the square to solve a quadratic equation?
2. What happens if the discriminant is negative in a quadratic equation?
3. Can you solve the quadratic equation by factoring?
4. How can the quadratic formula be used for equations with fractional coefficients?
5. How do you graph a quadratic equation to visually understand the solutions?

Tip: If the discriminant is positive, you get two real solutions; if it's zero, you get one real solution; and if it's negative, the solutions are complex numbers.