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The factored form of a quadratic equation is a way of expressing the equation as a product of its linear factors. For a quadratic equation like \(x^2 - bx + c = 0\), the factored form is typically \((x - r)(x - s) = 0\). Here, \(r\) and \(s\) are the roots, or solutions, to the equation.

In the context of the exercise, we were given that the sole solution is \(a\). Thus, both roots \(r\) and \(s\) equal \(a\), leading us to a factored form of \((x-a)(x-a)\). This indicates the quadratic equation has a repeated root, often referred to as a double root.

Using this form makes it simpler to understand the solutions of the equation and provides insight into its graph. The point \(x = a\) is the vertex of the parabola and is both a maximum or minimum point, depending on the orientation of the parabola.

The distributive property is a fundamental principle in algebra that allows us to expand expressions and equations. It states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) can be expanded to \(a \cdot b + a \cdot c\).

To use the distributive property on our quadratic equation \((x-a)(x-a)\), we apply this property to each term:

• Multiply \(x\) by each term in the second factor \((x-a)\): \(x \cdot x - x \cdot a\)
• Then multiply \(-a\) by each term in the second factor: \(-a \cdot x + a^2\)

This results in \(x^2 - a\cdot x - a\cdot x + a^2\). Combine like terms to finally form \(x^2 - 2a\cdot x + a^2\).

This step-by-step application of the distributive property simplifies a complex expression into a standard form quadratic equation.

The FOIL method is a specific application of the distributive property used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, representing the order in which you multiply the terms.

When expanding \((x-a)(x-a)\) using the FOIL method:

• First: Multiply the first terms in each binomial: \(x \cdot x = x^2\)
• Outer: Multiply the outer terms: \(x \cdot (-a) = -a \cdot x\)
• Inner: Multiply the inner terms: \(-a \cdot x = -a \cdot x\)
• Last: Multiply the last terms: \(-a \cdot -a = a^2\)

The resulting expression is \(x^2 - a \cdot x - a \cdot x + a^2\), which simplifies to \(x^2 - 2a \cdot x + a^2\).

Using the FOIL method helps students systematically calculate each part of the binomial product, preventing errors and ensuring accuracy in obtaining the expanded quadratic expression.