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A quadratic equation is an expression that equates to zero and is of the form \(ax^2 + bx + c = 0\). It is a polynomial equation of degree 2, meaning that the highest power of the variable \(x\) is 2. Quadratic equations are fundamental in algebra and appear in various mathematical problems and real-life applications.

When analyzing quadratic equations, the first step is to identify the coefficients \(a\), \(b\), and \(c\):

• \(a\) - the coefficient of \(x^2\)
• \(b\) - the coefficient of \(x\)
• \(c\) - the constant term

For instance, in the equation \(3x^2 + 5x - 8 = 0\), we have \(a = 3\), \(b = 5\), and \(c = -8\). Understanding these coefficients is crucial for solving the equation or determining its properties, such as its solutions.

The nature of solutions of a quadratic equation is determined by its discriminant, \(\Delta\). The discriminant is calculated using the formula \(\Delta = b^2 - 4ac\). This value helps us understand how many solutions exist and the type of solutions they are.

Here's what the discriminant reveals:

• If \(\Delta > 0\), there are two distinct real solutions. If it's a perfect square, the solutions are rational.
• If \(\Delta = 0\), there's exactly one real solution, which is also rational.
• If \(\Delta < 0\), there are no real solutions, only two complex solutions.

For the quadratic equation \(3x^2 + 5x - 8 = 0\), the discriminant is 121. Since 121 is greater than 0 and a perfect square, it indicates two distinct real rational solutions.

Rational solutions are solutions of an equation that can be expressed as a fraction of two integers. When dealing with quadratic equations, determining the rationality of solutions is crucial for understanding the nature of the equation's roots.

A quadratic equation has rational solutions when its discriminant is a perfect square. This is because the roots can be expressed using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). If \(\sqrt{b^2 - 4ac}\) is an integer, then the entire solution simplifies to a rational number.

In the example \(3x^2 + 5x - 8 = 0\), the discriminant is 121, which is \(11^2\), a perfect square. As a result, the equation has two rational solutions, making it easier to express these roots in a simple fractional form. Rational solutions are especially useful in contexts where precise, finite decimal representation is needed.