To find the solutions of the quadratic equation x² + 7x + 4, we can use the quadratic formula, which is given by:
x = (-b ± √(b² – 4ac)) / (2a)
In our equation, we identify the coefficients as follows:
- a = 1 (the coefficient of x²)
- b = 7 (the coefficient of x)
- c = 4 (the constant term)
Now, let’s follow these steps to calculate the discriminant (b² – 4ac):
b² – 4ac = 7² – 4(1)(4) = 49 – 16 = 33
Since the discriminant is positive (33 > 0), we will have two distinct real solutions. Now we can substitute the values back into the quadratic formula:
x = (–7 ± √33) / 2(1)
This simplifies to:
x = (–7 ± √33) / 2
Now we break this down into the two possible solutions:
- x₁ = (–7 + √33) / 2
- x₂ = (–7 – √33) / 2
Therefore, the solutions to the quadratic equation x² + 7x + 4 are:
- x₁ ≈ -1.42 (when using 7 + √33)
- x₂ ≈ -5.58 (when using 7 – √33)
In conclusion, the solutions are approximately x₁ ≈ -1.42 and x₂ ≈ -5.58.