The quadratic equation given is:

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

This equation can be solved using the quadratic formula, which is:

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

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

• $a = 4$
• $b = 4$
• $c = 1$

Step 1: Calculate the discriminant

The discriminant is the part under the square root in the quadratic formula:

$\Delta = b^2 - 4ac$

Substituting the values of $a$, $b$, and $c$:

$\Delta = 4^2 - 4(4)(1) = 16 - 16 = 0$

Step 2: Apply the quadratic formula

Since the discriminant is 0, there will be one real solution (a repeated root):

$x = \frac{-b \pm \sqrt{0}}{2a} = \frac{-4}{2(4)} = \frac{-4}{8} = -\frac{1}{2}$

Final Answer:

The solution to the equation $4x^2 + 4x + 1 = 0$ is:

$x = -\frac{1}{2}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the discriminant affect the nature of the roots in a quadratic equation?
2. Can every quadratic equation be solved using the quadratic formula?
3. How can you complete the square to solve $4x^2 + 4x + 1 = 0$?
4. What are the different methods for solving quadratic equations besides the quadratic formula?
5. How can you graphically represent the solution of $4x^2 + 4x + 1 = 0$?

Tip: Always check the discriminant first to determine the nature of the roots (real and distinct, real and repeated, or complex).