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The discriminant is a key component in solving quadratic equations, found inside the quadratic formula. It helps determine the nature of the roots of the equation. The discriminant is calculated using the formula: \[ D = b^2 - 4ac \]
In the problem, the quadratic equation given is: \[ x^2 + 4x + 6 = 0 \]
From this, we identify the coefficients: \[ a = 1 \] \[ b = 4 \] \[ c = 6 \]
Plugging these values into the discriminant formula we get: \[ D = 4^2 - 4(1)(6) \] This simplifies to: \[ D = 16 - 24 = -8 \]
Since the discriminant is negative (\( D < 0 \)), this means the quadratic equation has no real solutions and instead has complex solutions.

When the discriminant of a quadratic equation is negative, the solutions are not real numbers, but complex numbers. Complex numbers have a real part and an imaginary part.
The imaginary unit is denoted by \( i \), where \( i \) is defined as \( \sqrt{-1} \). This helps to understand complex solutions better. When solving the quadratic equation \[ x^2 + 4x + 6 = 0 \]
We found the discriminant \( D \) to be -8. Using this discriminant value in the quadratic formula, we see that it involves the square root of a negative number, leading to an imaginary number. Substituting \( D = -8 \) into the quadratic formula, we get: \[ x = \frac{-4 \pm \sqrt{-8}}{2(1)} \]
This breaks down as follows: \[ \sqrt{-8} = \sqrt{8( -1 )} = 2i\sqrt{2} \]
Therefore, the solutions are: \[ x = \frac{-4 \pm 2i\sqrt{2}}{2} = -2 \pm i\sqrt{2} \]
So the quadratic equation has two complex solutions: \[ x = -2 + i\sqrt{2} \] and \( x = -2 - i\sqrt{2} \)

The quadratic formula is a universal method for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is: \[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Where \( D = b^2 - 4ac \) is the discriminant. This formula is derived from completing the square of the general quadratic equation.
Let's apply the quadratic formula to our given equation: \[ x^2 + 4x + 6 = 0 \]
We have already identified the coefficients as \( a = 1, b = 4, c = 6 \)
Substitute these values into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{-8}}{2(1)} \]
Since the discriminant \( D \) is -8, we get: \[ x = \frac{-4 \pm 2i\sqrt{2}}{2} = -2 \pm i\sqrt{2} \]
This demonstrates how to find the complex solutions of a quadratic equation using the quadratic formula. The quadratic formula is very powerful, as it works for all quadratic equations, whether the solutions are real or complex.