To solve the quadratic equation x² + 5x + 3 = 0, we can use the quadratic formula, which is given by:
x = (-b ± √(b² – 4ac)) / 2a
In our equation, the coefficients are:
- a = 1 (the coefficient of x²)
- b = 5 (the coefficient of x)
- c = 3 (the constant term)
Now, we will substitute these values into the quadratic formula. First, we need to calculate the value of the discriminant (b² – 4ac):
b² = 5² = 25
4ac = 4 * 1 * 3 = 12
Now, calculate the discriminant:
b² – 4ac = 25 – 12 = 13
Since the discriminant is positive, we will have two real and distinct solutions. Now, we can plug the values back into the quadratic formula:
x = (-5 ± √13) / (2 * 1)
This simplifies to:
x = (-5 ± √13) / 2
Now, we can express the two solutions:
- x₁ = (-5 + √13) / 2
- x₂ = (-5 – √13) / 2
These represent the two values of x that satisfy the original quadratic equation. To summarize:
- The solutions are approximately:
- x₁ ≈ -0.30
- x₂ ≈ -4.70
This completes the solution of the quadratic equation x² + 5x + 3 = 0.