Of course, I can help you solve this quadratic equation using the quadratic formula!
The quadratic formula is a formula that gives you the solutions of a quadratic equation in the form of ax^2 + bx + c = 0, where a, b, and c are coefficients. In this case, we have the equation x^2 + 4x + 6 = 0.
The quadratic formula states that the solutions (or roots) of the quadratic equation can be found using the following formula:
x = (-b ± √(b^2 - 4ac)) / (2a),
where ± represents taking both the positive and negative roots, and √(b^2 - 4ac) represents the square root of the discriminant.
Now let's apply the quadratic formula to solve the given equation:
a = 1, b = 4, c = 6.
Substituting these values into the formula:
x = (-4 ± √(4^2 - 4*1*6)) / (2*1),
Simplifying the expression further:
x = (-4 ± √(16 - 24)) / 2,
x = (-4 ± √(-8)) / 2.
Now we have a square root of a negative number, which means we will be dealing with complex-number solutions.
To simplify the square root of -8, we can rewrite it as the square root of -1 times the square root of 8. The square root of -1 is denoted by the imaginary unit "i". The square root of 8 can be written as 2√2.
Therefore, we have:
x = (-4 ± 2√2i) / 2,
Simplifying further:
x = -2 ± √2i.
This gives us the two complex-number solutions for the given equation as:
x = -2 + √2i, and
x = -2 - √2i.
And that's how you can use the quadratic formula to solve the quadratic equation x^2 + 4x + 6 = 0 and find the complex-number solutions. Make sure to double-check all the steps and calculations, especially when dealing with complex numbers. Good luck with your practice and upcoming test!