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The discriminant is a key component in the quadratic formula that helps determine the nature of the roots of a quadratic equation. It is represented by the expression \( b^2 - 4ac \) and is derived from the general form of a quadratic equation: \( ax^2 + bx + c = 0 \). The value of the discriminant gives insight into the possible types of solutions:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root (also called a repeated root).
• If the discriminant is negative, there are no real roots, but two complex conjugate roots.

In our problem, the equation is \( x^2 + 4x + 2 = 0 \). By substituting \( a = 1 \), \( b = 4 \), and \( c = 2 \) into the discriminant formula, we calculated \( 4^2 - 4 \times 1 \times 2 \), which results in a discriminant of 8.
This positive value indicates that our quadratic equation has two distinct real roots.

A quadratic equation is a type of polynomial equation of degree 2. It has the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The quadratic equation can be solved using several methods, such as factoring, graphing, completing the square, or using the quadratic formula.
The quadratic formula is a reliable method used to find the roots of any quadratic equation, whether it can be easily factored or not. The formula is given by:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
In the original problem, we used the quadratic formula to solve \( x^2 + 4x + 2 = 0 \). By identifying the coefficients \( a = 1 \), \( b = 4 \), and \( c = 2 \), we substituted them into the formula to find the solution.
Using the discriminant helps in predicting the nature of the roots, as explained earlier, and makes the quadratic formula a powerful tool in solving quadratic equations systematically.

The roots of a polynomial are the values of \( x \) that make the polynomial equal to zero. In the context of a quadratic equation, these roots are the solutions that satisfy the equation \( ax^2 + bx + c = 0 \). Whether these roots are real or complex, they are critical for understanding the behavior of the polynomial.
When calculating the roots of a quadratic equation using the quadratic formula, we find the points where the parabola (the graph of the quadratic equation) intersects the x-axis.
In our problem, using the quadratic formula, we calculated the roots of the equation \( x^2 + 4x + 2 = 0 \). The solutions obtained were \( x = -2 + \sqrt{2} \) and \( x = -2 - \sqrt{2} \), indicating where the parabola crosses the x-axis.

• These roots are essential because they denote the solutions to the equation.
• Finding them can also provide insight into real-world problems modeled by quadratic equations.

Understanding how the roots relate to the graph of the polynomial helps in visualizing and interpreting solutions.