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).

Learn how to solve the quadratic equation 4x² + 4x + 1 = 0 step-by-step using the quadratic formula. This problem demonstrates the use of the discriminant and provides a solution suitable for students in Grades 9-10.

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