To solve the quadratic equation x² + 2x + 4 = 0 using the quadratic formula, we first identify the coefficients in the standard form of the equation, which is ax² + bx + c = 0. Here:
- a = 1 (coefficient of x²)
- b = 2 (coefficient of x)
- c = 4 (constant term)
The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / (2a)
Now, we can substitute the values of a, b, and c into the formula:
x = (-(2) ± √((2)² - 4(1)(4))) / (2(1))This simplifies to:
x = (-2 ± √(4 - 16)) / 2Calculating the discriminant (b² – 4ac):
4 - 16 = -12Since the discriminant is negative (-12), this indicates that the solutions will be complex (imaginary) numbers. Now, we proceed with the square root of -12:
√(-12) = √(12) * √(-1) = 2√3iThus, we can now substitute this back into our equation:
x = (-2 ± 2√3i) / 2Simplifying this gives:
x = -1 ± √3iTherefore, the solutions to the equation x² + 2x + 4 = 0 are:
- x = -1 + √3i
- x = -1 – √3i
In conclusion, the quadratic formula has provided us with complex solutions, which often occur in quadratic equations with a negative discriminant. Feel free to reach out if you have further questions about quadratic equations or complex numbers!