factor each trinomial completely and explain how you got it
9k^2 + 66k + 21
5x^2 - 18 + 9
4x^2 - 35x + 49
1. For the trinomial 9k^2 + 66k + 21, we first look for two numbers that multiply to the product of the coefficient of the squared term and the constant term, which is 9 * 21 = 189, and add up to the coefficient of the middle term, which is 66. The numbers that satisfy these conditions are 9 and 21.
So, we can rewrite the trinomial as 9k^2 + 63k + 3k + 21.
Now, we can factor by grouping:
9k(k + 7) + 3(k + 7)
= (9k + 3)(k + 7)
= 3(3k + 1)(k + 7)
2. For the trinomial 5x^2 - 18x + 9, we first look for two numbers that multiply to the product of the coefficient of the squared term and the constant term, which is 5 * 9 = 45, and add up to the coefficient of the middle term, which is -18. The numbers that satisfy these conditions are -15 and -3.
So, we can rewrite the trinomial as 5x^2 - 15x - 3x + 9.
Now, we can factor by grouping:
5x(x - 3) - 3(x - 3)
= (5x - 3)(x - 3)
= (x - 3)(5x - 3)
3. For the trinomial 4x^2 - 35x + 49, we first look for two numbers that multiply to the product of the coefficient of the squared term and the constant term, which is 4 * 49 = 196, and add up to the coefficient of the middle term, which is -35. The numbers that satisfy these conditions are -28 and -7.
So, we can rewrite the trinomial as 4x^2 - 28x - 7x + 49.
Now, we can factor by grouping:
4x(x - 7) - 7(x - 7)
= (4x - 7)(x - 7)
= (x - 7)(4x - 7)
Tell me about fractions, and why it is difficult. Why
do you think that is? Share some tips or tricks for classmates who may be struggling with fractions.
Fractions are a mathematical concept that represent a part of a whole. They consist of a numerator (top part) and a denominator (bottom part), separated by a fraction bar. Fractions can be used to represent quantities that are not whole numbers, like 1/2, 3/4, or 5/6.
Fractions can be difficult for some people because they require a good understanding of the relationship between parts and the whole, as well as basic arithmetic operations like addition, subtraction, multiplication, and division. Some common challenges with fractions include:
1. Adding and subtracting fractions with different denominators.
2. Multiplying and dividing fractions.
3. Converting between fractions and decimals.
4. Simplifying fractions to their lowest terms.
One reason fractions can be difficult is that they involve multiple steps and operations, which can be overwhelming for some students. Additionally, fractions require a strong grasp of foundational math concepts, such as division and multiplication, which some students may struggle with.
Here are some tips and tricks for classmates who may be struggling with fractions:
1. Practice regularly with fractions to build familiarity and confidence.
2. Understand the relationship between fractions, decimals, and percentages.
3. Use visual aids, such as fraction bars or circles, to help visualize fractions.
4. Simplify fractions by dividing the numerator and denominator by their greatest common factor.
5. Learn the rules for adding, subtracting, multiplying, and dividing fractions.
By consistently practicing and understanding the basic principles of fractions, students can improve their skills and feel more confident when working with fractions. Remember that it's okay to ask for help from teachers, tutors, or classmates if you're struggling with fractions.