Question
Can you help me understand the difference of when to use as a example
(x^2 + x) Or (x+x)(x-x) or (x-x)(x+x) or (x-x)(x-x) etc.
(x^2 + x) Or (x+x)(x-x) or (x-x)(x+x) or (x-x)(x-x) etc.
Answers
Answered by
GPT 3.5
Certainly! Let's break down each expression and understand when and how to use them as examples:
1. (x^2 + x): This expression represents the sum of x squared and x. It can be used as an example when you want to simplify or factor a quadratic expression. For example, if you have an equation like x^2 + 5x + 6, you can use the expression x^2 + x to demonstrate factoring by grouping.
2. (x + x)(x - x): This expression represents the product of x added to x, multiplied by x subtracted by x. The result of this expression will be 0, since x - x is always equal to 0. This can be used as an example to demonstrate the zero product property, which states that if the product of two or more factors is equal to 0, then at least one of the factors must be 0.
3. (x - x)(x + x): This expression represents the product of x subtracted by x, multiplied by x added to x. Similar to the previous example, the result of this expression will be 0, as x - x is equal to 0. This can also be used to illustrate the zero product property.
4. (x - x)(x - x): This expression represents the product of x subtracted by x, multiplied by x subtracted by x. The result of this expression will be x^2 - 2x + x^2, which simplifies to 2x^2. This can be used as an example to demonstrate multiplying binomials and simplifying the resulting expression.
I hope this helps clarify the differences between these expressions and when to use them as examples! Let me know if you have any other questions.
1. (x^2 + x): This expression represents the sum of x squared and x. It can be used as an example when you want to simplify or factor a quadratic expression. For example, if you have an equation like x^2 + 5x + 6, you can use the expression x^2 + x to demonstrate factoring by grouping.
2. (x + x)(x - x): This expression represents the product of x added to x, multiplied by x subtracted by x. The result of this expression will be 0, since x - x is always equal to 0. This can be used as an example to demonstrate the zero product property, which states that if the product of two or more factors is equal to 0, then at least one of the factors must be 0.
3. (x - x)(x + x): This expression represents the product of x subtracted by x, multiplied by x added to x. Similar to the previous example, the result of this expression will be 0, as x - x is equal to 0. This can also be used to illustrate the zero product property.
4. (x - x)(x - x): This expression represents the product of x subtracted by x, multiplied by x subtracted by x. The result of this expression will be x^2 - 2x + x^2, which simplifies to 2x^2. This can be used as an example to demonstrate multiplying binomials and simplifying the resulting expression.
I hope this helps clarify the differences between these expressions and when to use them as examples! Let me know if you have any other questions.
Answer
I'm confused with these two problems
9x^2 - 64 = (3x -8)(3x +8)
9x^2 + 12x + 4 = (3x + 2)^2
On the first problem why can't it be written
(3x - 8)^2 the same as the second problem
9x^2 - 64 = (3x -8)(3x +8)
9x^2 + 12x + 4 = (3x + 2)^2
On the first problem why can't it be written
(3x - 8)^2 the same as the second problem
Answered by
GPT 3.5
I see where the confusion lies. Let's break down each problem to clarify their differences:
1. 9x^2 - 64 = (3x - 8)(3x + 8): In this problem, we are looking to factor the binomial expression 9x^2 - 64. You correctly factored it into the difference of squares (3x - 8)(3x + 8), because 9x^2 is the same as (3x)^2 and 64 is the same as 8^2. This can't be written as (3x - 8)^2 because it is not a perfect square trinomial and does not fall into the special case of a square of a binomial.
2. 9x^2 + 12x + 4 = (3x + 2)^2: In this problem, we have a perfect square trinomial, 9x^2 + 12x + 4, which can be factored as a square of a binomial, (3x + 2)^2. This is possible because 9x^2 is the same as (3x)^2, 12x is double the product of the coefficients of the square's terms (2 * 3 * 3x), and 4 is the same as 2^2.
In summary, the first problem involves factoring the difference of squares, while the second problem involves factoring a perfect square trinomial. It's important to recognize the specific patterns and properties of each expression in order to correctly factor them. I hope this clarifies the differences between the two problems! Let me know if you need further assistance.
1. 9x^2 - 64 = (3x - 8)(3x + 8): In this problem, we are looking to factor the binomial expression 9x^2 - 64. You correctly factored it into the difference of squares (3x - 8)(3x + 8), because 9x^2 is the same as (3x)^2 and 64 is the same as 8^2. This can't be written as (3x - 8)^2 because it is not a perfect square trinomial and does not fall into the special case of a square of a binomial.
2. 9x^2 + 12x + 4 = (3x + 2)^2: In this problem, we have a perfect square trinomial, 9x^2 + 12x + 4, which can be factored as a square of a binomial, (3x + 2)^2. This is possible because 9x^2 is the same as (3x)^2, 12x is double the product of the coefficients of the square's terms (2 * 3 * 3x), and 4 is the same as 2^2.
In summary, the first problem involves factoring the difference of squares, while the second problem involves factoring a perfect square trinomial. It's important to recognize the specific patterns and properties of each expression in order to correctly factor them. I hope this clarifies the differences between the two problems! Let me know if you need further assistance.