Factoring a Perfect Square Trinomial

Q1. Factor the perfect square trinomial: $4x^2 - 28x + 49$

Background

Topic: Factoring Perfect Square Trinomials

This question tests your ability to recognize and factor a trinomial that is a perfect square. Perfect square trinomials have the form $a^2 - 2ab + b^2$ or $a^2 + 2ab + b^2$, and can be factored into $(a - b)^2$ or $(a + b)^2$ respectively.

Key Terms and Formulas

• Perfect Square Trinomial: A trinomial that can be written as the square of a binomial.

• Factoring Formula:

  • $a^2 - 2ab + b^2 = (a - b)^2$

  • $a^2 + 2ab + b^2 = (a + b)^2$

Step-by-Step Guidance

1. Identify the first and last terms: $4x^2$ and $49$. Notice that $4x^2$ is $(2x)^2$ and $49$is $7^2$.

2. Check the middle term: $-28x$. For a perfect square trinomial, the middle term should be $-2ab$ where $a = 2x$ and $b = 7$.

3. Calculate $-2ab$: $-2 \times 2x \times 7 = -28x$. This matches the middle term, confirming it is a perfect square trinomial.

4. Write the trinomial in the form $(a - b)^2$ using $a = 2x$ and $b = 7$.

Try solving on your own before revealing the answer!