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. Recognize that the quadratic expression is in the form \(x^2 - 6x + 9\), which is a trinomial that might be a perfect square.
2. Recall the perfect square trinomial formula: \((a - b)^2 = a^2 - 2ab + b^2\).
3. Identify \(a\) and \(b\) such that \(a^2 = x^2\) and \(b^2 = 9\), so \(a = x\) and \(b = 3\).
4. Check if the middle term \(-6x\) matches \(-2ab = -2 \times x \times 3 = -6x\), which it does.
5. Write the factored form as \((x - 3)^2\).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratics involves rewriting a quadratic expression as a product of two binomials. This process helps simplify expressions and solve equations. Recognizing patterns like perfect square trinomials or using methods such as factoring by grouping is essential.

Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial, typically in the form a^2 ± 2ab + b^2 = (a ± b)^2. Identifying this pattern allows quick factoring without trial and error.

Using the Quadratic Formula or Completing the Square

When factoring is not straightforward, the quadratic formula or completing the square can find roots of the quadratic. These roots help express the quadratic as a product of linear factors, aiding in factoring and solving equations.