Solve 4x² + 4x + 1 = 0
Solve the quadratic equation 4x² + 4x + 1 = 0 using the quadratic formula.
Solution:
Discriminant: Δ = 0 (1 repeated root)
Step-by-Step Solution
Step 1
Identify the Standard Form
A quadratic equation has the form: ax² + bx + c = 0
From the given equation:
• a = 4 (coefficient of x²)
• b = 4 (coefficient of x)
• c = 1 (constant term)
Standard form: 4x² + 4x + 1 = 0
4x² + 4x + 1 = 0
Step 2
Calculate the Discriminant
The discriminant determines the nature of the roots.
D = b² - 4ac
D = (4)² - 4(4)(1)
D = 16 - 16
D = 0
Since D = 0, the equation has exactly ONE repeated root.
D = 0
Step 3
Solve by Factoring (AC Method)
For 4x² + 4x + 1 = 0:
Step 1: Find ac = 4 × 1 = 4
Step 2: Find factors of 4 that sum to 4
The equation factors to:
(4x + 2)(x + 0.5) = 0
\text{AC Method}
Step 4
Find the Solutions
Setting each factor to zero:
x₁ = -0.5
x₂ = -0.5
x_1 = -0.5, x_2 = -0.5
Step 5
Verify the Solutions
Substitute each solution back into the original equation:
For x = -0.5:
4(-0.5)² + 4(-0.5) + 1
= 0 ✓
For x = -0.5:
4(-0.5)² + 4(-0.5) + 1
= 0 ✓
\text{Both solutions verified!}
Step 6
Final Answer
x = -0.5 (repeated root)