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A quadratic equation is a type of polynomial equation that involves a variable raised to the second power. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). This means that the highest exponent of the variable \( x \) is 2. Quadratic equations are fundamental in algebra and appear often in various real-world applications.

• The term \( ax^2 \) is the quadratic term.
• The term \( bx \) is the linear term.
• The term \( c \) is the constant term.

Understanding the form and structure of quadratic equations is crucial for solving them using different methods.

The discriminant is a key component of the quadratic formula and helps us determine the nature of the solutions to a quadratic equation. It is found under the square root in the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The discriminant is the expression \( b^2 - 4ac \).
The value of the discriminant tells us:

• If \( b^2 - 4ac > 0 \), there are two distinct real solutions.
• If \( b^2 - 4ac = 0 \), there is exactly one real solution (the solutions are repeated).
• If \( b^2 - 4ac < 0 \), there are two complex (not real) solutions.

In our equation \( x^2 + 6x + 3 = 0 \), we calculated the discriminant as 24. Since 24 is greater than zero, we have two distinct real solutions.

The solutions of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These solutions can be found using several methods, including factoring, completing the square, or using the quadratic formula.
Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), provides a direct way of finding solutions, especially when the equation doesn't factor easily. In our example, substituting \( a = 1 \), \( b = 6 \), \( c = 3 \) into the quadratic formula gives us:\[ x = \frac{-6 \pm \sqrt{24}}{2} \]
After simplifying, the solutions are:

• \( x = -3 + \sqrt{6} \)
• \( x = -3 - \sqrt{6} \)

These solutions indicate the points where the quadratic equation would cross or touch the \( x \)-axis if graphed.

Simplifying radicals is an important step in solving quadratic equations when dealing with the square root part of the quadratic formula. It involves breaking down a radical expression into its simplest form.
For instance, with the discriminant \( \sqrt{24} \), we simplify as follows:

• The number 24 can be factored into \( 4 \times 6 \).
• Since \( \sqrt{4} = 2 \), the expression becomes \( 2\sqrt{6} \).

By simplifying radicals, we make calculations cleaner and easier to interpret. In our equation, simplifying \( \sqrt{24} \) into \( 2\sqrt{6} \) allowed us to further simplify each component of the quadratic formula, resulting in the final solutions: \( x = -3 \pm \sqrt{6} \).