To solve the quadratic equation 2x² + 4x + 7 = 0 using the quadratic formula, we first need to identify the coefficients of the equation. The standard form of a quadratic equation is:
ax² + bx + c = 0
For our equation:
- a = 2
- b = 4
- c = 7
Next, we will use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
Now, let’s calculate the discriminant (the part under the square root):
b² – 4ac = (4)² – 4(2)(7)
Calculating further:
- 16 – 56 = -40
Since the discriminant is negative (-40), this means there are no real solutions, but we do have complex solutions.
Now, we can simply substitute back into our formula:
x = (-4 ± √(-40)) / (2 * 2)
Now, let’s simplify it:
- √(-40) = √(40) * √(-1) = √(40)i
- √(40) = √(4 * 10) = 2√(10)
Now substituting this back in:
x = (-4 ± 2√(10)i) / 4
This simplifies to:
- x = -1 ± (√(10)/2)i
Thus, we conclude:
The solutions to the equation 2x² + 4x + 7 = 0 are:
x = -1 + (√(10)/2)i and x = -1 – (√(10)/2)i
These solutions are complex numbers, indicating that the parabola represented by this equation does not intersect the x-axis.