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In the realm of algebra, a quadratic equation is any equation that can be manipulated into the standard form 'ax² + bx + c = 0', where 'a', 'b', and 'c' are constants, and 'a' is not zero. The quadratic equation is foundational in mathematics because it is one of the simplest polynomial equations to solve. This type of equation typically has two solutions or "roots" which represent the points where the graph of the quadratic function intersects the x-axis.

To find these roots algebraically, you can use several methods:

• Factoring, which involves expressing the quadratic as a product of its linear factors.
• The Quadratic Formula, given by \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a},\]which provides a solution for any quadratic equation.
• Completing the Square, which makes the quadratic expression a perfect square trinomial, simplifying the solving process.

Understanding how to manipulate and solve quadratic equations is vital when working with polynomial functions, especially when aiming to find their zeros or intercepts, as shown in the given exercise.

Factoring is a technique used to break down equations into simpler components. It's like reverse distributing, where you express a polynomial as a product of its simplest factors. For quadratic equations, this means finding two linear expressions that multiply together to give the original quadratic expression. This approach was utilized in the exercise to solve the polynomial function algebraically.

In the example of \[ f(x)=9x^{4}-25x^{2}, \]we performed factoring by initially rewriting it in terms of a new variable, \(X = x^2\). The equation transformed into a quadratic form \[ 9X^{2}-25X = 0. \]Here, we factor out the common term \(X\), resulting in \[ X(9X - 25) = 0. \]This indicates that the polynomial may be broken into two factors, implying that each can be zero, thus determining the roots \(X = 0\) and \(X = \frac{25}{9}\). This highlights the power of factoring in simplifying the problem of finding polynomial zeros.

Algebraic Manipulation involves rewriting expressions or equations in algebra to isolate variables, simplify expressions, or solve equations. This might involve operations such as addition, subtraction, multiplication, division, or even substitution, as demonstrated in the example exercise.

In the exercise given,\[ f(x)=9x^{4}-25x^{2}, \]algebraic manipulation was a key step. By substituting \(X = x^2\), the problem was simplified into a more familiar quadratic form, which could then be easily factored and solved. This step of substitution is a powerful technique because it can transform a complex polynomial into a solvable equation by temporarily changing the variable.

After solving for \(X\), further manipulation was needed. Re-substituting \(X = x^2\) yielded equations:

• \(x^2 = 0\), giving \(x = 0\).
• \(x^2 = \frac{25}{9}\), providing \(x = \pm \frac{5}{3}\).

Through algebraic manipulation, you can find the zeros of polynomial equations, fully utilizing your problem-solving skills in algebra.