Factor by any method. See Examples 1–7. 4z²+28z+49

Verified step by step guidance

1. Identify the quadratic expression to factor: \$4z^2 + 28z + 49$.
2. Check if the quadratic is a perfect square trinomial by comparing it to the form \(a^2 + 2ab + b^2\).
3. Find \(a\) and \(b\) such that \(a^2 = 4z^2\) and \(b^2 = 49\). Here, \(a = 2z\) and \(b = 7\).
4. Verify if the middle term \$28z$ equals \(2ab = 2 \times 2z \times 7 = 28z\), confirming it is a perfect square trinomial.
5. Write the factored form as \((2z + 7)^2\).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratics involves rewriting a quadratic expression as a product of two binomials or other factors. This process simplifies expressions and solves 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 a² + 2ab + b² = (a + b)². Identifying this pattern helps factor expressions quickly and accurately.

Greatest Common Factor (GCF)

The GCF is the largest factor shared by all terms in an expression. Factoring out the GCF simplifies the expression and can make further factoring easier. Always check for a GCF before applying other factoring methods.