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The discriminant is a crucial component in solving quadratic equations. It is the part of the quadratic formula under the square root, expressed as \( b^2 - 4ac \). This value, often called \( \Delta \), helps us determine the nature of the roots of a quadratic equation. By examining \( \Delta \), you can instantly know if the equation has real or complex solutions.

When solving the quadratic equation \( 2x^2 - 6x + 1 = 0 \), we identified a, b, and c as \( 2, -6, \) and \( 1 \) respectively. Plugging these values into the discriminant formula gave us \( \Delta = 28 \). This is greater than zero, indicating that there are two distinct real solutions.

The relationship between the discriminant and the nature of the roots is straightforward:

• If \( \Delta > 0 \), the equation has two distinct real solutions.
• If \( \Delta = 0 \), the equation has exactly one real solution, with a repeated root or multiplicity of two.
• If \( \Delta < 0 \), the equation has two nonreal, complex solutions.

This information simplifies solving quadratic equations by providing insight without needing to immediately solve for actual values.

Real solutions of a quadratic equation refer to the values of \( x \) that satisfy the equation and are real numbers (not involving imaginary units). In the given problem, the discriminant \( \Delta = 28 \) indicated that our equation \( 2x^2 - 6x + 1 = 0 \) possesses two distinct real solutions.

This means when we solve the equation, both solutions will be real numbers, which are easy to plot on a standard number line.

Understanding the nature of solutions helps in predicting the type of graph the quadratic equation will produce. For distinct real solutions:

• The graph of the equation will intersect the x-axis at two separate points, corresponding to the two real solutions.
• This indicates a parabola opening upwards or downwards, depending on the sign of coefficient \( a \).

Recognizing whether solutions will be real or complex is essential when assessing the behavior of quadratic functions in various applications, from physics to finance.

The quadratic formula provides a reliable method for finding the roots of a quadratic equation. This formula is especially useful when factoring is tricky or the solutions are non-integer or irrational.

The formula itself is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Substituting values from our problem \( a = 2 \), \( b = -6 \), and \( \Delta = 28 \) (from our discriminant calculation), gives us:\[ x = \frac{6 \pm \sqrt{28}}{4} \]
This simplifies to:\[x = \frac{3 \pm \sqrt{7}}{2}\]resulting in two real solutions: \( x = \frac{3 + \sqrt{7}}{2} \) and \( x = \frac{3 - \sqrt{7}}{2} \).

Steps using the quadratic formula:

• Calculate the discriminant \( b^2 - 4ac \) for insight into the nature of the roots.
• Substitute the values of \( a \), \( b \), and the discriminant into the formula.
• Solve for \( x \) to find the roots which the equation equates to zero.

This method is foundational in algebra and is a powerful tool, enabling clear and systematic solutions to quadratic equations where other methods might fail.