To solve the quadratic equation 3x² + 5x + 1 = 0 using the quadratic formula, we start by identifying the coefficients in the standard form of a quadratic equation: ax² + bx + c = 0.
In this equation:
- a = 3
- b = 5
- c = 1
The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / (2a)
Now, let’s plug in the values of a, b, and c into the formula.
1. **Calculate the discriminant**:
b² – 4ac = 5² – 4(3)(1) = 25 – 12 = 13
2. **Substitute into the quadratic formula**:
x = (–5 ± √13) / (2 × 3)
x = (–5 ± √13) / 6
3. **Calculate the two possible solutions**:
x₁ = (–5 + √13) / 6
x₂ = (–5 – √13) / 6
Thus, the solutions for the quadratic equation 3x² + 5x + 1 = 0 are:
- x₁ ≈ –0.434
- x₂ ≈ –1.566
These are the roots of the equation, found using the quadratic formula!