To solve the quadratic equation 3x² + 5x + 1 = 0, we can use the quadratic formula, which is given by:
x = (-b ± √(b² – 4ac)) / 2a
In this formula, a, b, and c represent the coefficients of the quadratic equation ax² + bx + c = 0. For our equation, we have:
- a = 3
- b = 5
- c = 1
Now, we will calculate the discriminant, D = b² – 4ac:
D = 5² – 4(3)(1) = 25 – 12 = 13
Since the discriminant is positive (D > 0), we will have two distinct real roots. Now we can substitute the values of a, b, and D into the quadratic formula:
x = (-5 ± √13) / (2 * 3)
Let’s calculate the two possible values for x:
1. For the positive root:
x₁ = (-5 + √13) / 6
2. For the negative root:
x₂ = (-5 – √13) / 6
These two expressions represent the two solutions for the quadratic equation 3x² + 5x + 1 = 0.