91Ó°ÊÓ

A quadratic equation is a type of polynomial equation of degree 2, meaning the highest power of the variable is 2. It is commonly written in the standard form: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
Quadratic equations are fundamental in algebra because they frequently appear in different topics and have multiple applications in real-world problems and other mathematical concepts. Quadratic equations can have:

• **Two distinct solutions**: When the discriminant \(b^2 - 4ac\) is positive, the equation has two distinct real roots.
• **One solution**: When the discriminant is zero, there is one real repeated root.
• **No real solutions**: If the discriminant is negative, the equation has two complex roots.

Understanding the structure of quadratic equations helps in identifying patterns for factoring and predicting the number of solutions they may have.

The roots or solutions of a polynomial are the values of the variable that make the polynomial equal to zero. In the context of quadratic equations, these roots are the solutions that satisfy the equation \(ax^2 + bx + c = 0\).
For the given problem, which features a fourth-degree polynomial, the process involves identifying quadratic expressions within it, like \(x^2\), to solve for its roots. The roots can be found using methods like factorization or the quadratic formula: \[ x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a} \] For instance, in this problem, we identify the expressions \((x^2 - 1)\) and \((x^2 - 1.25)\). Solving these gives us the roots of the polynomial.
Once the roots are known, they provide insight into the behavior of the polynomial and can serve as a guide to graphing the polynomial or understanding its intersections with the x-axis. Roots are a fundamental concept as they reveal information about the function’s zero points, especially in polynomial functions.

Factoring is the process of breaking down a complex expression into simpler ones that, when multiplied together, produce the original. It is an essential algebraic technique, especially when dealing with polynomials.
For quadratic and higher-order polynomials, factoring begins by looking for any common factors among terms. If none exist, like in the problem \(4x^4 - 9x^2 + 5\), we can then rewrite the expression to find hidden quadratic forms, facilitating complete factorization. Key factoring techniques include:

• **Factoring by grouping**: Arranging terms to find common products among groups.
• **Using the difference of squares**: Applying the identity \(a^2 - b^2 = (a - b)(a + b)\).
• **Factoring trinomials**: Rewriting expressions as products of two binomials when possible.

In our specific exercise, the expression \(x^2 - 1\) is further factorable using difference of squares into \((x - 1)(x + 1)\). Utilizing these techniques simplifies the polynomial, making it easier to solve and understand how it behaves graphically and algebraically.