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Quadratic equations are mathematical statements that take the form of \( ax^2 + bx + c = 0 \). They represent equations where the highest power of the unknown variable \( x \) is two. In this exercise, our given quadratic equation is \( x^2 + 4x + 2 = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients which help define the parabola associated with the equation. For our particular example:

• \( a = 1 \)
• \( b = 4 \)
• \( c = 2 \)

These coefficients can determine many interesting properties about the graph and its roots. Quadratic equations like these commonly appear in various practical applications, ranging from physics to finance. Learning how to handle them gives a solid foundation for tackling more advanced algebraic topics.

The discriminant is an essential part of the quadratic formula, which helps us understand the nature of the roots of the equation. It is given by the expression \( b^2 - 4ac \) within the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For our equation \( x^2 + 4x + 2 = 0 \), we calculate the discriminant using:

• \( b = 4 \)
• \( a = 1 \)
• \( c = 2 \)

By substituting these values, we get:\[ 4^2 - 4 \times 1 \times 2 = 16 - 8 = 8 \]A positive discriminant like \( 8 \) indicates that the quadratic equation has two distinct real solutions. Understanding the discriminant can quickly inform you about the number of solutions without needing to solve the entire equation.

Exploring the nature of roots tells us about the characteristics of the solutions of a quadratic equation. The nature of roots depends entirely on the value of the discriminant \( b^2 - 4ac \):

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root, often called a repeated or double root.
• If the discriminant is negative, there are no real roots; instead, the roots are complex or imaginary.

Since our equation \( x^2 + 4x + 2 = 0 \) has a positive discriminant of \( 8 \), we conclude it possesses two distinct real roots. Knowing the nature of roots is crucial for predicting the behavior of the solutions in the context of a problem.

Real roots of a quadratic equation are the solutions that are real numbers, not imaginary or complex numbers. For our equation \( x^2 + 4x + 2 = 0 \), the quadratic formula gives us the roots:\[ x = \frac{-4 \pm 2\sqrt{2}}{2} \]By simplifying, we find:

• First root: \( x_1 = -2 + \sqrt{2} \)
• Second root: \( x_2 = -2 - \sqrt{2} \)

These real roots can be verified by substituting back into the original equation to ensure they satisfy it. Real roots mean that these solutions correspond to the points where the graph of the quadratic equation intersects the x-axis. Understanding real roots helps to visualize the solutions and their practical implications in real-life scenarios.