How can I use the quadratic formula to solve the equation 2x² + 4x + 7 = 0 and what are the possible values for x?

To solve the quadratic equation 2x² + 4x + 7 = 0, we will utilize the quadratic formula, which is given by:

x = (-b ± √(b² – 4ac)) / (2a)

In our equation, we identify the coefficients as follows:

Now, we will substitute these values into the quadratic formula:

x = ( -4 ± √(4² - 4(2)(7)) ) / (2(2))

This simplifies to:

x = ( -4 ± √(16 - 56) ) / 4

Calculating the discriminant:

16 - 56 = -40

Since the discriminant is negative (-40), this indicates that the solutions for x will be complex numbers. We continue the calculation:

x = ( -4 ± √(-40) ) / 4

Next, we rewrite √(-40) as:

√(-40) = √(40) * √(-1) = √(40)i

Now, remember that √(40) can be simplified:

√(40) = √(4 * 10) = 2√(10)

Substituting this back into our equation:

x = ( -4 ± 2√(10)i ) / 4

Now, let’s separate the two terms:

x = -1 ± (√(10)i) / 2

Thus, we find the values of x:

x = -1 + (√(10)i) / 2

x = -1 - (√(10)i) / 2

In conclusion, the solutions to the equation 2x² + 4x + 7 = 0 are: