Example 4: Factoring quadratics as a perfect square of a difference: (a-b)^2 | Khan Academy

TL;DR

Learn how to factor perfect square trinomials by applying the pattern of squaring binomials with a leading coefficient that is not 1.

Transcript

Factor 25x squared minus 30x plus 9. And we have a leading coefficient that's not a 1, and it doesn't look like there are any common factors. Both 25 and 30 are divisible by 5, but 9 isn't divisible by 5. We could factor this by grouping. But if we look a little bit more carefully here, see something interesting. 25 is a perfect square, and 25x squ... Read More

Key Insights

• ❎ Factoring perfect square trinomials involves recognizing perfect squares and applying the pattern of squaring binomials.
• 😃 The leading coefficient of the trinomial affects the values of 'a' and 'b' in the factored form.
• ❎ The product of 'a' and 'b' must be negative to achieve the middle term of the trinomial.
• 🧑‍🏭 Different factorizations can exist due to the ability to factor out a common negative 1.
• 😑 Understanding the process of factoring perfect square trinomials is essential for simplifying expressions and solving equations.
• ❓ Recognizing patterns and applying them can simplify the factoring process.
• 💯 Perfect square trinomials have a distinctive structure that can be identified through knowledge of perfect squares.

Questions & Answers

Q: How can you factor a perfect square trinomial?

To factor a perfect square trinomial, identify if it has a leading coefficient not equal to 1 and look for perfect squares as the first and last term. Apply the pattern of squaring binomials to factor the trinomial.

Q: What is the pattern for squaring binomials?

When you square a binomial ax + b, the result is a^2x^2 + 2abx + b^2. This pattern shows that squaring a binomial involves multiplying the two terms, the first term squared, and the second term squared.

Q: How do you determine the values for factoring a perfect square trinomial?

The perfect square trinomial can be factored as (ax + b)^2 or (ax - b)^2, where:

• 'a' is the square root of the first term.
• 'b' is the square root of the last term.
• The signs of 'a' and 'b' are opposite to achieve the desired product.

Q: Can a perfect square trinomial be factored in different ways?

Yes, a perfect square trinomial can be factored in multiple ways. It can be factored as (ax + b)^2 or (ax - b)^2, depending on the signs of 'a' and 'b'. Both factorizations result in the same expanded form, as the binomial can be factored out a common negative 1.

Summary & Key Takeaways

• The provided content explains how to factor a perfect square trinomial.

• The process involves recognizing perfect squares and applying the pattern of squaring binomials.

• The coefficients of the perfect square trinomial determine the values to be used in factoring.