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The discriminant is a crucial tool when dealing with quadratic equations. It helps us determine the nature of the roots, or solutions, for these equations. In a quadratic equation, which is typically in the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is represented by the formula \(b^2 - 4ac\). This value can tell us a lot about the solutions:

• When \(\Delta > 0\), the quadratic equation has two distinct real solutions.
• When \(\Delta = 0\), there is exactly one real solution, sometimes referred to as a repeated or double root.
• When \(\Delta < 0\), the quadratic equation has no real solutions but instead has two complex solutions.

Understanding these outcomes allows us to predict whether solving the equation will lead to real values or not.

Real solutions of a quadratic equation occur when we substitute the coefficients \(a, b, \) and \(c\) into the quadratic formula. However, the discriminant plays a pivotal role as it indicates the reality of these solutions before any further calculation.When discussing real solutions, they can be geometrically visualized as the points where the parabola intersects the x-axis in a graph. If a quadratic equation has two real solutions, it means the parabola touches the x-axis at two points. Conversely, one real solution means the vertex of the parabola just touches the x-axis.In our exercise, we want to find the values of \(a\) that result in no real solutions. We use the condition \(\Delta < 0\), ensuring that the parabola stays above or below the x-axis without crossing it.

Inequalities are essential when determining the conditions for the discriminant. To explore when a quadratic equation has no real solutions, we set up an inequality. Given a quadratic equation like \(ax^2 - 6x + 2 = 0\), we calculate using the discriminant \((-6)^2 - 4(a)(2) < 0\).Our simplified inequality is \(36 - 8a < 0\). Solving this inequality involves isolating \(a\) to find the interval where the statement holds true:

• Subtract 36 from both sides: \(-8a < -36\).
• Divide by -8, remembering to reverse the inequality sign, leading to \(a > 4.5\).

This inequality shows us that for values of \(a\) greater than 4.5, the quadratic equation will have no real solutions. This logical deduction helps us analyze and choose correct values in exercises.