Factoring Quadratics: Step-by-Step Solutions
In this article, we'll walk through factoring two quadratic expressions. Factoring quadratics is a fundamental skill in algebra, and mastering it will help you solve equations, simplify expressions, and understand more advanced mathematical concepts. We'll break down each problem step-by-step, so you can follow along and improve your factoring skills.
1. Factoring $25 n^2 - 30 n + 9$
Our goal is to factor the quadratic expression $25 n^2 - 30 n + 9$. When factoring quadratics, the first thing we look for is whether the expression is a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. The general form of a perfect square trinomial is $a^2 \pm 2ab + b^2$, which factors into $(a \pm b)^2$. Let's see if our expression fits this pattern.
In our expression, $25 n^2$ can be written as $(5n)^2$, and $9$ can be written as $3^2$. So, we can think of $a$ as $5n$ and $b$ as $3$. Now, we need to check if the middle term, $-30n$, matches $2ab$ or $-2ab$. In this case, we have:
Since our middle term is $-30n$, it matches $-2ab$. Therefore, the expression $25 n^2 - 30 n + 9$ is indeed a perfect square trinomial and can be factored as $(a - b)^2$, where $a = 5n$ and $b = 3$.
So, we have:
To verify this, let's expand $(5n - 3)^2$:
This matches our original expression, so our factoring is correct. Therefore, the factored form of $25 n^2 - 30 n + 9$ is $(5n - 3)^2$.
Therefore, the correct answer is B. $(5 n-3)^2$
In summary, recognizing the perfect square trinomial pattern is crucial for efficiently factoring these types of expressions. By identifying 'a' and 'b' and verifying the middle term, you can quickly factor the quadratic into its squared binomial form. This approach not only saves time but also reduces the chances of making errors during the factoring process. Always double-check your factored form by expanding it to ensure it matches the original quadratic expression, reinforcing your understanding and accuracy.
2. Factoring $4 b^2 + 20 b + 25$
Now, let's factor the quadratic expression $4 b^2 + 20 b + 25$. Similar to the previous problem, we'll check if this expression is a perfect square trinomial. Recall that a perfect square trinomial has the form $a^2 \pm 2ab + b^2$, which factors into $(a \pm b)^2$.
In our expression, $4 b^2$ can be written as $(2b)^2$, and $25$ can be written as $5^2$. So, we can consider $a$ as $2b$ and $b$ as $5$. Now, we need to check if the middle term, $20b$, matches $2ab$:
Since our middle term is $20b$, it matches $2ab$. Therefore, the expression $4 b^2 + 20 b + 25$ is a perfect square trinomial and can be factored as $(a + b)^2$, where $a = 2b$ and $b = 5$.
So, we have:
To verify this, let's expand $(2b + 5)^2$:
This matches our original expression, so our factoring is correct. Therefore, the factored form of $4 b^2 + 20 b + 25$ is $(2b + 5)^2$.
Therefore, the correct answer is C. $(2 b+5)^2$
In summary, the process of identifying and factoring perfect square trinomials relies on recognizing the pattern $a^2 + 2ab + b^2$. By expressing the first and last terms as squares and verifying that the middle term is twice the product of the square roots of the first and last terms, you can confidently factor the trinomial into $(a + b)^2$. Always expand the factored form to confirm it matches the original expression, ensuring accuracy and solidifying your understanding of factoring techniques.
In conclusion, mastering factoring techniques, especially recognizing perfect square trinomials, is essential for algebraic problem-solving. By understanding the underlying patterns and practicing regularly, you can enhance your ability to factor quadratic expressions quickly and accurately. Remember to always verify your factored forms to ensure they match the original expressions.
For further learning on factoring and other algebraic concepts, visit Khan Academy's Algebra Section. This resource offers comprehensive lessons, practice exercises, and videos to help you deepen your understanding of algebra.
