91Ó°ÊÓ

The factored form of a quadratic equation is a vital concept that allows us to express the equation as a product of its linear factors. This form is particularly useful when you know the roots or solutions of the equation. For any quadratic equation with solutions, say \( p \) and \( q \), the equation can be expressed in factored form as:

• \((x - p)(x - q) = 0\)

This means that the quadratic equation is the result of multiplying two linear expressions.
For example, if a quadratic equation has solutions \( 7 \) and \(-7\), the factored form would be \((x - 7)(x + 7) = 0\). This step of rewriting the equation is crucial for further simplification or solving purposes.
Factored form clearly shows the roots and indicates where the graph of the quadratic function intersects the x-axis.

The difference of squares is an algebraic technique that simplifies the multiplication of two binomials that follow a pattern. It takes the form:

• \((a - b)(a + b) = a^2 - b^2\)

This formula allows us to easily multiply these binomials without expanding them fully.
It is especially useful in quadratic equations, such as when converting a factored form like \((x - 7)(x + 7)\) to the simplified expression of \(x^2 - 49\).
Here, \(a = x\) and \(b = 7\), so applying the difference of squares gives us \(x^2 - 49\), a simpler and equivalent expression.

The standard form of a quadratic equation is an essential structure that makes these equations easy to analyze and solve. It is written as:

• \(ax^2 + bx + c = 0\)

In this form, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The term \(ax^2\) is the quadratic term, \(bx\) is the linear term, and \(c\) is the constant term.
Understanding this format is crucial because it provides a clear pathway to apply solving methods like factoring, completing the square, or using the quadratic formula.
In the case where the quadratic equation has been solved to give the expression \(x^2 - 49\), this is already in standard form with \(a = 1\), \(b = 0\), and \(c = -49\). Observing standard form helps in determining the nature of the roots based on the discriminant \(b^2 - 4ac\).