Exercises 100–102 will help you prepare for the material covered in the next section. Factor: x^2 - 6x + 9

Verified step by step guidance

1. Start by identifying the quadratic expression: \(x^2 - 6x + 9\).
2. Notice that the expression is a perfect square trinomial. A perfect square trinomial is of the form \((a - b)^2 = a^2 - 2ab + b^2\).
3. Compare \(x^2 - 6x + 9\) with \(a^2 - 2ab + b^2\) to identify \(a\) and \(b\). Here, \(a = x\) and \(b = 3\) because \(2ab = 6x\) implies \(2 \cdot x \cdot 3 = 6x\).
4. Rewrite the expression as \((x - 3)^2\) since it matches the form \((a - b)^2\).
5. Verify by expanding \((x - 3)^2\) to ensure it equals \(x^2 - 6x + 9\).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic in the form ax^2 + bx + c as a product of two binomials. This process is essential for simplifying expressions and solving equations. The goal is to express the quadratic in a form that reveals its roots or solutions, which can be found using the zero-product property.

The Zero-Product Property

The zero-product property states that if the product of two factors equals zero, then at least one of the factors must be zero. This principle is crucial when solving quadratic equations after factoring, as it allows us to set each factor equal to zero to find the solutions of the equation.

Perfect Square Trinomials

A perfect square trinomial is a specific type of quadratic expression that can be factored into the square of a binomial. The general form is a^2 ± 2ab + b^2, which factors to (a ± b)^2. Recognizing perfect square trinomials, like x^2 - 6x + 9, simplifies the factoring process and helps in identifying the roots quickly.